Quasi-invariant and pseudo-differentiable measures in Banach spaces

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Q UASI - INVARIANT AND P SEUDO - DIFFERENTIABLE M EASURES IN BANACH S PACES S ERGEY V. L UDKOVSKY

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Contents Preface

vii

Acknowledgement

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Notation

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1 1 3 28 57 67 73 86

Non-Archimedean Valued Measures 2.1. Introduction . . . . . . . . . . . . . . . . 2.2. Non-Archimedean Valued Distributions . 2.3. Quasi-invariant Ks -Valued Measures . . . 2.4. Pseudo-differentiable Ks -Valued Measures 2.5. Convergence of Ks -Valued Measures . . . 2.6. Measures with Particular Properties . . . 2.7. Comments . . . . . . . . . . . . . . . . .

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91 91 92 109 121 124 128 140

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Algebras of Real Measures on Groups 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Algebras of Measures and Functions . . . . . . . . . . . . . . . . . . . . . 3.3. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 143 152

4

Algebras of Non-Archimedean Measures on Groups 153 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2. Algebras of Measures and Functions . . . . . . . . . . . . . . . . . . . . . 153 4.3. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2

Real-Valued Measures 1.1. Introduction . . . . . . . . . . . . . . . 1.2. Distributions and Families of Measures 1.3. Quasi-invariant Measures . . . . . . . . 1.4. Pseudo-differentiable Measures . . . . . 1.5. Convergence of Measures . . . . . . . . 1.6. Measures with Particular Properties . . 1.7. Comments . . . . . . . . . . . . . . . .

A Operators in Banach Spaces

173

v

vi

Contents

B Non-Archimedean Polyhedral Expansions 179 B.1. Ultra-uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.2. Polyhedral Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 References

191

Index

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Preface This book is devoted to new results of investigations of non-Archimedean functional analysis, which is becoming more important nowadays due to the development of nonArchimedean mathematical physics, particularly, quantum mechanics, quantum field theory, theory of super-strings and supergravity [VV89, VVZ94, ADV88, Cas02, DD00, Ish84, Khr90, Lud99t, Lud03b, Mil84, Jan98]. Recently non-Archimedean analysis was found to be useful in dynamical systems, mathematical biology, mathematical psychology, cryptology and information theory. On the other hand, quantum mechanics is based on measure theory and probability theory. The results of this book published mainly in papers [Lud02a, Lud03s2, Lud04a, Lud96c, Lud99a, Lud00a, Lud99t, Lud01f, Lud00f, Lud99s, Lud04b] have served for investigations of non-Archimedean stochastic processes [Lud0321, Lud0341, Lud0348, Lud01f, LK02]. Stochastic approach in quantum field theory is actively used and investigated especially in recent years (see, for example, and references therein [AHKMT93, AHKT84]). As it is well-known in the theory of functions great role is played by continuous functions and differentiable functions. In the classical measure theory the analog of continuity is quasi-invariance relative to shifts and actions of linear or non-linear operators in the Banach space. Moreover, differentiability of measures is the stronger condition and there is very large theory about it in the classical case. Apart from it the non-Archimedean case was less studied. Since there are not differentiable functions from the p-adic field Qp into R or into another p′ -adic non-Archimedean field Q p′ with p 6= p′ , then instead of differentiability of measures their pseudo-differentiability is considered. Traditional or classical mathematical analysis and functional analysis work mainly over the real and complex fields. But there are well-known many other infinite fields with non-trivial multiplicative norms since the end of the 19-th century. If a multiplicative norm in a field K or a norm in a vector space X over K satisfies instead of the triangle inequality stronger condition: |x + y| ≤ max(|x|, |y|) for each x, y ∈ K or in X respectively, then it is called the non-Archimedean norm. Such fields and vector spaces with non-Archimedean norms are frequently called for short non-Archimedean fields and nonArchimedean normed spaces correspondingly. Therefore, mathematical analysis and functional analysis over non-Archimedean fields develop already during rather long period of time, but they remain substantially less elaborated in comparison with that of the classical one. The first chapter of this book is devoted to real-valued measures and in the second chapter measures with values in non-Archimedean fields are described. Though the results of these two chapters have served in investigations of quasi-invariant and pseudo-differentiable

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Sergey V. Ludkovsky

measures on topological totally disconnected groups which can be non-locally compact such as Lie groups, diffeomorphism groups and geometric wrap or loop groups of Banach manifolds over non-Archimedean fields. They also were used for investigations of representations of such groups in the series of papers [Lud99a, Lud00a, Lud99t, Lud98b, Lud01s, Lud02b, Lud0348, Lud08] (for comparison, in the case of non-locally compact groups over R or C see, for example, also [VGG75, PS68, AHKMT93, AHKT84, Kos94, Shim94, Lud99r, Lud01f]). But restricting by the scope of this book only few results illustrating applications of such measures are given in Chapters 3 and 4, which provide main differences between the case of locally compact groups and non-locally compact groups. Quasi-invariance and pseudo-differentiability of measures can also be used for studying properties of transition conditional measures of stochastic processes, for solution of pseudodifferential or anti-derivational stochastic equations. In the case of locally compact groups there is possible to construct a nontrivial Haar measure on a group and this serves to define C∗ -algebra corresponding to this group. This is the crucial point in investigations of their representations in linear spaces and finding invariant closed linear subspaces [Nai68, FD88]. In particular, decomposition of unitary representations in complex Hilbert spaces into a direct integral of topologically irreducible representations can be accomplished with the help of this technique. In the case of non-locally compact groups there does not exist any nontrivial Haar measure, but only a measure quasi-invariant relative to left (or right) shifts by elements of a proper subgroup. Then it is possible to associate with the latter measure an algebra over such group, but it is not the C∗ -algebra, its structure is much more complicated, though for it there is proved the analog of the Gelfand-Naimark theorem in Chapter 3 for real-valued measures and its non-Archimedean counterpart for non-Archimedean valued measures in Chapter 4. Effective ways to use quasi-invariant and pseudo-differentiable measures are given in the cited above articles of the author. Professor I.V. Volovich had been discussing with me the matter and interested in results of my investigations of non-Archimedean analogs of Gaussian measures such as to satisfy as many Gaussian properties as possible as he has planned to use such measures in non-Archimedean quantum field theory. The question was not so simple. He has supposed that properties with mean values, moments, projections, distributions and convolutions of such measures can be considered analogously. This matter we had been debating with Professor B. Diarra, who had doubted that all Gaussian circumstances can be fulfilled. But thorough analysis has shown, that not all properties can be satisfied, because in such case the linear space would have a structure of the R-linear space (see §I.6 and §II.6). Nevertheless, many of the properties there is possible to satisfy in the non-Archimedean case also. Gaussian measures are convenient to work in the classical case, but in the non-Archimedean case they do not play so great role. Strictly speaking no any nontrivial Gaussian measure exists in the non-Archimedean case, but measures having few properties analogous to that of Gaussian can be outlined. Supplying them with definite properties depends on a subsequent task for which problems they may be useful. For example, if each projection µY of a measure µ on a finite dimensional subspace Y over a field K is equivalent to the Haar measure λY on Y , then this is very well property. But in the classical case, as it is well-known, such property does not imply that the measure µ is Gaussian, since each measure νY (dx) = f (x)λY (dx) with

Preface

ix

f ∈ L1 (Y, λY , R) is absolutely continuous relative to the Lebesgue measure λY on Y and this does not imply Gaussian properties of moments or its characteristic functional (see, for example, [GV61, DF91, HT74, VTC85]). The class of measures having such properties of projections is much wider than that of Gaussian and is described by the Kolmogorov and Kakutani theorems. This book is devoted to general theory of quasi-invariant and pseudo-differentiable measures not restricting the theory by a particular class of measures. Earlier versions of the results about quasi-invariant and pseudo-differentiable measures were communicated to A.C.M. van Rooij and W.H. Schikhof (Nijmegen) in 1994-1995 and also to M. van der Put (Groningen), who were interested in it. This text was thoroughly read by B. Diarra (Clermont-Ferrand) who has recommended to write about real-valued measures and measures with values in non-Archimedean fields separately, because their theory differ substantially. Results of these investigations were also communicated at research seminars and lectures of Mathematical Department of Blaise Pascal University in Clermont-Ferrand and at Department of Mathematical Physics of Steklov Mathematical Institute in Moscow and at Chair of Higher Algebra at Mathematical Department of Moscow State University (M.V. Lomonosov), at the conference by non-Archimedean analysis at V¨axj¨o University (Sweden). The starting point for this work was specific non-Archimedean general measure theory of A.P. Monna and T.A. Springer, A.C.M. van Rooij and W.H. Schikhof, etc. Also some results of investigations of V.S. Vladimirov, I.V. Volovich, E.I. Zelenov and A.Yu. Khrennikov were used, who considered it for problems of non-Archimedean quantum mechanics. More detailed discussions of sources and somewhat related works are given in comments and introductions to each chapter. This book is devoted to more specific measure theory of quasi-invariant and pseudodifferentiable measures in Banach spaces including infinite-dimensional over fields. The author has written this theory in details, though it also opens ways for further investigations in this new area. The results of this book provide also wider classes of quasi-invariant and pseudo-differentiable measures on non-locally compact groups and non-Archimedean manifolds with the help of approaches described in papers on groups and manifolds cited above. In the first chapter quasi-invariant and pseudo-differentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial normalization are defined and constructed and studied. Measures are considered with values in R. Theorems and criteria are formulated and proved about quasi-invariance and pseudodifferentiability of measures relative to linear and non-linear operators on X. Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are proved. Convolutions of measures and infinite products of measures also are considered. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated. In the second chapter measures are considered with values in non-Archimedean fields, for example, the field Qp of p-adic numbers. Classes of quasi-invariant and pseudodifferentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial normalization are described and studied. Their quasi-invariance and pseudo-differentiability relative to linear operators and non-linear transformations on

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Sergey V. Ludkovsky

X is investigated. The corresponding theorems are demonstrated. The important instrument for analysis of the measure is its characteristic functional. Therefore, characteristic functionals of a measure are studied below. Other important cornerstones are the nonArchimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems, which are investigated as well. Moreover, infinite products of measures are considered and the analog of the Kakutani theorem is proved. Then spaces of measures are studied and a convergence of nets of quasi-invariant and pseudo-differentiable measures is investigated. In the third and the fourth chapters properties of quasi-invariant measures (real and non-Archimedean valued respectively) relative to dense subgroups on topological groups are considered. There predominantly non-locally compact groups are major objects such as (i) a group of diffeomorphisms Di f f (t, M) of non-Archimedean manifold M in cases of locally compact and non-locally compact M, where t is a class of smoothness, (ii) a general Banach-Lie group over a classical or non-Archimedean field, (iii) wrap or loop groups of real and non-Archimedean manifolds. The main feature there consists in that algebras of quasi-invariant measures are introduced and studied. Generally these algebras appear to be non-commutative and nonassociative over non-locally compact groups. Therefore, they do not induce any C∗ -algebra and the Gelfand-Mazur theorem is not valid already for such algebras. Nevertheless, for them analogs of the Gelfand-Mazur theorem are proved. One may mention that the nonlocal compactness causes a twisted algebraic structure of measure spaces. This situation can be compared with representation theory of groups in non-Archimedean linear spaces. Over infinite fields with non-trivial non-Archimedean multiplicative norms the aforementioned theorem is not accomplished due to existence of transcendental extensions of such fields (see [Roo78] and references therein). For reading of this book it is better to have some basic knowledge of the material contained in works [Roo78, Sch84, DF91, Eng86, FD88], though below all necessary definitions and notations are given. The present book can be used for studying of this part of functional analysis, for reference and for further investigations. Its results can be used not only in Mathematics in functional analysis, theory of topological and Lie groups and their representations, topological algebras, dynamical systems, probability theory, random functions and stochastic processes and equations, integral transforms, but also in quantum mechanics and quantum field theory, mathematical biology, mathematical psychology, cryptology, information theory, etc.

Acknowledgement The author is sincerely grateful to all colleagues with whom this work or its parts were discussed: B. Diarra, A. Escassut, I.V. Volovich, A. Khrennikov, A. H. Bikulov, S. V. Kozyrev, O. G. Smolyanov. Moscow August 2008 S ERGEY V. L UDKOVSKY

Notation

A f (X, µ) §I.2.1, II.2.1; Bco(X) §II.2.1; B f (X) §I.2.1; B(X, x, r) §I.2.2; B+ , C+ §I.2.29; c0 (α, K), PL §I.2.2; C(X, K) §I.2.6; ˆ Γ), τ(Y ) §I.2.26; C(Y, C(Y, Γ), τ(Y ) §II.2.20; Cs Introduction of Chapter II; c0 ({Hi : i ∈ No }) §4.17; δ0 §I.2.8; Fp Introduction of Chapter I; Fp (θ) Introduction of Chapter I; K Introduction of Chapter I; Ks Introduction of Chapter II; L(X, µ, Ks ) §II.2.4; l2 ({Hi : i ∈ No }) §3.16; M(X) §I.2.1, II.2.1; Mt (X) §I.2.1, II.2.1; µL §I.2.2, §II.2.2; {µLn : n} §I.2.2, §II.2.2; µ1 ∗ µ2 §I.2.11, §II.2.8; ν ≪ µ, ν ∼ µ, ν ⊥ µ §I.2.36, II.2.31; Qp Introduction of Chapter I; Qs Introduction of Chapter II; Us Introduction of Chapter II and §II.4.1; ψq,µ §I.2.14; θ(z) = µˆ §I.2.6, §II.2.5; χξ §I.2.6, II.2.5.

Chapter 1

Real-Valued Measures 1.1.

Introduction

There are known works on integration in a Banach space over the field R of real numbers or the field C of complex numbers [Bou63-69, Chr74, Con84, DF91, Sko74, VTC85]. But for a non-Archimedean Banach space X, which is over a field with a non-Archimedean multiplicative norm, this theory is less developed. Integration theory in X is a very important part of the non-Archimedean analysis. The period is such that the advances of quantum mechanics and different branches of modern physics related, for example, with theories of elementary particles lead to the necessity of developing integration theory in a non-Archimedean Banach space [ADV88, DD00, Ish84, Mil84, VVZ94, Jan98]. Certainly, it may also be useful for the development of non-Archimedean functional analysis. As it is well-known non-Archimedean analysis develops rapidly in recent years and has many principal differences from the classical analysis [Khr90, Roo78, Sch84, Sch89, Sch71, VVZ94]. Linear topological spaces over non-Archimedean fields are totally disconnected. Therefore, classes of smoothness for functions and compact operators are defined for them quite differently from that of the classical case. In such spaces also the notion of the orthogonality of vectors has obtained quite another meaning. We mention also that in the non-Archimedean case the Radon-Nikodym theorem and the Lebesgue theorem about convergence are not true in the classical form, but their analogs are true under more rigorous and another conditions. Very strong differences are for measures with values in non-Archimedean fields in comparison with that of with real- or complex-valued, because classical notions of σ-additivity and quasi-invariance have lost their meaning. Nonetheless, the development of the non-Archimedean functional analysis and its applications in non-Archimedean quantum mechanics [VVZ94, Khr90, Jan98] leads to the necessity of solving such problems. Frequently advances of quantum mechanics on manifolds and quantum field theory are related with diffeomorphism groups and wrap or loop groups, their representations and measures on them [Ish84, Mil84, Lud99t, Lud00a]. In publications [Lud96, Lud98s, Lud99t, Lud98b, Lud00a] quasi-invariant measures on diffeomorphism and wrap or loop groups and also on manifolds were constructed. Then such measures were used for the investigation of unitary including irreducible representations in complex Hilbert spaces of topological and Lie groups [Lud99t, Lud98b, Lud99a]. The theorems demonstrated in this book enlarge classes of measures on such groups and man-

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ifolds. As the consequence this also enlarges classes of available representations. For example, theorems analogous to that of the Minlos-Sazonov type characterize measures with the help of characteristic functionals and compact operators. Compact operators are more useful in the non-Archimedean case, rather than nuclear operators in the classical case. Of exceptional importance are theorems of the Bochner-Kolmogorov and Kakutani type characterizing products of measures and their absolute continuity relative to others measures. In this chapter measures are considered on Banach spaces, though the results given below can be developed for more general topological vector spaces, for example, it is possible to follow the ideas of works [Mad91c, Mad91a, Mad85], in which were considered non-Archimedean analogs of the Minlos-Sazonov theorems for real-valued measures on topological vector spaces over non-Archimedean fields of zero characteristic. But it is impossible to make in one chapter or book. In this book, apart from articles of Ma¸drecki, measures are considered also with values in non-Archimedean fields (see Chapter 2). For the cases of real-valued measures also Banach spaces over non-Archimedean fields K of characteristic char(K) > 0 are considered. Recall that a real-valued measure m on a locally compact Hausdorff totally disconnected Abelian topological group G is called the Haar measure, if (H) m(x + A) = m(A) for each x ∈ G and each Borel subset A in G. It is useful to start from measures on the field K equivalent to the Haar measure up to a multiplier which is a measurable function on K. Many of definitions and theorems described below are the non-Archimedean analogs of classical results. But frequently their formulations and proofs differ strongly. The reader, need not be familiar with the part of classical functional analysis about measures in infinitedimensional spaces, because this book contains all necessary definitions and results. Below in §2 sequences of weak distributions, characteristic functions of measures and their properties are defined and investigated. The non-Archimedean analogs of the MinlosSazonov and Bochner-Kolmogorov theorems are presented. For this quasi-measures also are considered, because they are tightly related with measures. In §3 products of measures are described together with their density functions. In the present chapter broad classes of quasi-invariant measures are defined and constructed. Further theorems about quasiinvariance of measures under definite linear and non-linear transformations U : X → X are demonstrated. §4 contains a notion of pseudo-differentiability of measures. This is very important, because for functions f : K → R there is not any notion of differentiability, that is there is not such non-linear non-trivial function f . Then criteria for the pseudodifferentiability are studied. In §5 theorems about converegence of measures there are given with taking into account their quasi-invariance and pseudo-differentiability, that is, in the corresponding spaces of measures. The main results are Theorems 2.27, 2.35, 3.4, 3.20, 3.24, 3.25, 4.2, 4.3, 4.5, 4.7, 5.7-5.10. The first chapter tackles real-valued measures. Below in the second chapter measures with values in non-Archimedean fields are considered as well. This is caused by differences in definitions, formulations of statements and their proofs in two such principally distinct cases. To avoid repeating when differences are small only briefly matters are discussed in the second chapter and refereing the first. Notations. Henceforth, K denotes a locally compact infinite field with a non-trivial norm, then the Banach space X is over K. In the present chapter measures on X have values

Real-Valued Measures

3

in the real field R. We assume that either K is a finite algebraic extension of the field of p-adic numbers Qp or char(K) = p. In the latter case K is isomorphic with the field Fp (θ) of formal power series consisting of elements x = ∑∞j=N a j θ j , where a j ∈ Fp , |θ| = p−1 , Fp is the finite field of p elements, p is a prime number, N = N(x) ∈ N, N denotes the set of natural numbers N = {1, 2, 3, . . . }. If x 6= 0, aN 6= 0, then |x| = p−N , certainly |0| = 0. Mention that Qp is of zero characteristic char(Qp ) = 0. Recall that each non-zero p-adic number x can be written in the form x = ∑∞j=N x j p j , where x j ∈ {0, 1, . . . , p − 1} for each j, while xN 6= 0 and N = N(x) ∈ N. The norm of such x is |x| := p−N , while |0| = 0. An equivalent multiplicative norm is |x|c with a marked positive constant c > 0. These imply that the locally compact field K has the Haar measures with values in R [Roo78]. If X is a Hausdorff topological space with a small inductive dimension ind(X) = 0 (see [Eng86]), then E denotes an algebra of subsets of X. As a rule we use E ⊃ B f (X) for real-valued measures, where B f (X) is a Borel σ-field of X in §2.1, A f (X, µ) is the completion of E by a measure µ in §2.1. Remind that a topological space X is called zero-dimensional, if X is a non-void T1 space having a base of topology consisting of clopen subsets, where a subset U in X is called clopen if it is closed and open simultaneously. Then each zero-dimensional space is regular (Tychonoff) space. For any subset A of a topological space X a boundary of A is defined as Fr A := cl(A) ∩ cl(X \ A) = [cl(A)] \ Int(A), where cl(A) denotes the closure of A in X and Int(A) is the interior of A in X. Suppose that X is a regular space and n is a non-negative integer, 0 ≤ n ∈ Z. Put / (MU1) ind(X) = −1 if and only if X = 0; (MU2) ind(X) ≤ n if for each point x ∈ X and every its neighborhood V an open subset U ⊂ X exists such that x ∈ U ⊂ V and ind(Fr U) ≤ n − 1; (MU3) ind(X) = n if ind(X) ≤ n and the inequality ind(X) ≤ n − 1 is not satisfied; (MU4) ind(X) = ∞ if the inequality ind(X) ≤ n is not accomplished for any n. The number ind(X) is called the small inductive dimension of the topological space X.

1.2.

Distributions and Families of Measures

2.1. For a Hausdorff topological space X with a small inductive dimension ind(X) = 0 [Eng86] the Borel σ-field we denote by B f (X), where B f stands as the abbreviation from two words. Henceforth, measures µ are given on a measurable space (X, E), where E is a σ-algebra of subsets in X. The completion of E relative to µ we denote by A f (X, µ). The total variation of a measure µ with values in R on a subset A is denoted by kµ|A k or |µ|(A) for A ∈ A f (X, µ). Recall that |µ|(A) := sup ∑ |µ(E)| π E∈π

it is first defined on E and then is extended on the completion for each A ∈ A f (X, µ), where π is an arbitrary finite partition of a set A. If µ is non-negative and µ(X) = 1, then it is called a probability measure.

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Sergey V. Ludkovsky

A measure µ on E is called Radon, if for each ε > 0 there exists a compact subset C ⊂ X such that kµ|(X\C) k < ε. Henceforth, M(X) denotes a space of norm-bounded measures kµk = |µ|(X) < ∞, Mt (X) is its subspace of Radon norm-bounded measures. Remind also the following facts. I. Definitions. A family of subsets {Sn : n ∈ ϒ} in a set X is called shrinking, if for each k, n ∈ ϒ there exists m ∈ ϒ so that Sm ⊂ Sk ∩ Sn , where ϒ is some set. An ultra-metric space (X, d) is called spherically complete, if every shrinking sequence {Bn : n ∈ N} of balls in X has a non-void intersection. By a spherical completion < F, T > of a normed K-vector space E the pair < F, T > is so called consisting of a spherically complete space F and a K-linear isometry T : E → F such that F has not a proper spherically complete linear subspace containing in itself T (E). The spherical completion is frequently denoted simply also by F instead of < F, T >. For a normed K-linear space E two vectors x, y ∈ E are called orthogonal, if kax+byk = max(kaxk, kbyk) for all a, b ∈ K. For a real number 0 < t ≤ 1 a finite or an infinite sequence of elements x j ∈ E is called t-orthogonal, if ka1 x1 + · · · + am xm + · · · k ≥ t max(ka1 x1 k, . . . , kam xm k, . . . ) for each a1 , . . . , am , · · · ∈ K with a1 x1 + · · · + am xm + · · · ∈ E. We say, that a Banach space E over a field K complete relative to its non-Archimedean multiplicative norm has the countable type, if E is a closed K-linear span of some countable its subset. If Z is a K-linear subspace of the normed space E over K, then E is called the immediate extension of Z, if 0 is an unique element in E orthogonal to the subspace Z. Let ω be a non-void set and there is given a function s : ω → (0, ∞). For a function f : ω → K put k f ks := sup{| f (x)|s(x) : x ∈ ω}. The set of all functions f : ω → K, for which the norm k f ks is finite, forms the K-vector space denoted by l ∞ (K, ω; s), which is the Banach space relative to the norm k ∗ ks . By c0 (K, ω; s) we denote the closed subspace in l ∞ (K, ω; s), consisting from all functions f , satisfying the additional condition: for each b > 0 the set {x ∈ ω : | f (x)|s(x) > b} is finite. If a function s takes only one fixed value, for example, the unit, then we omit s from the notation of the Banach space. Further we shall need two theorems. II. Theorem (5.13 [Roo78]). For each Banach space E over the field K complete relative to its non-Archimedean multiplicative norm the following conditions are equivalent: (α). Each maximal orthogonal system of elements in E is a basis. (β). Each closed K-linear subspace in E has an orthogonal complement. (γ). Each closed K-linear subspace in E of countable type has an orthogonal complement. (δ). E is not an immediate extension of of any its proper closed K-linear subspace in E. (ε). Each countable orthogonal system of elements in E can be extended up to an orthogonal basis in E. III. Theorem (5.16 [Roo78]. For an infinite-dimensional Banach space E over the field K complete relative to its non-Archimedean multiplicative norm conditions (α − ε) from Theorem II are equivalent with any of the following conditions (ζ − ι): (ζ). Each closed K-linear subspace in E is spherically complete.

Real-Valued Measures

5

(η). E has an orthogonal basis and is spherically complete. (θ). Each strictly decreasing sequence of values of a norm in E converges to zero. (ι). The normalization group ΓK := {|x| : x ∈ K, x 6= 0} of the field K is discrete in (0, ∞). For a number π ∈ K so that 0 < |π| < 1 and |π| is the generator of the normalization group ΓK of the field K there exists a set ω and a function s : ω → (|π|, 1] so that E is isometrically K-linearly isomorphic with c0 (K, ω; s), where the set of values of a function s is well ordered. If E is infinite-dimensional and of countable type over K, then conditions (α − ι) are equivalent to the spherical completeness of the Banach space E. 2.2. As usually denote by c0 (α, K) the Banach space c0 (α, K) := {x : x = (x j : j ∈ α), card( j : |x j |K > b) < ℵ0 for each b > 0}, where α is a set considered as an ordinal due to the Kuratowski-Zorn lemma, card(A) denotes the cardinality of A, the norm is kxk := sup(|x j | : j ∈ α). A dimension of X over K is by the definition dimK X := card(α). Each Banach space X over K in view of Theorems 5.13 and 5.16 [Roo78] is K-linearly topologically isomorphic with c0 (α, K), since the field K is spherically complete. For each closed Klinear subspace L in X there exists an operator of a projection PL : X → L. Moreover, an orthonormal in the non-Archimedean sense basis in L has a completion to an orthonormal basis in X such that PL can be defined in accordance with a chosen basis. If A ∈ B f (L), then PL−1 (A) is called a cylindrical subset in X with a base A, BL := −1 PL (B f (L)), B0 := ∪(BL : L ⊂ X, L is a Banach subspace ,dimK X < ℵ0 ). The minimal σalgebra σB0 generated by B0 coincides with B f (X), if dimK X ≤ ℵ0 . Henceforward, it is assumed that α ≤ ω0 , where ω0 is the initial ordinal with the cardinality ℵ0 := card(N). This implies that there exists an increasing sequence of Banach subspaces Ln ⊂ Ln+1 ⊂ · · · such that cl(∪[Ln : n]) = X, dimK Ln = κn for each n, where cl(A) = A¯ denotes the closure of A in X for A ⊂ X. We fix a family of projections PLLnm : Lm → Ln so that PLLnm PLLkn = PLLkm for all m ≥ n ≥ k. Projections of the measure µ onto linear subspaces L denoted by µL (A) := µ(PL−1 (A)) for each A ∈ B f (L) compose the consistent family: µLn (A) = µLm (PL−1 (A) ∩ Lm ) n

(1)

for each m ≥ n, since there are projectors PLLnm , where κn ≤ ℵ0 and there may be chosen κn < ℵ0 for each n. An arbitrary family of measures {µLn : n ∈ N} having property (1) is called a sequence of weak distributions (see also [DF91, Sko74] in the classical case). By B(X, x, r) we denote the ball {y : y ∈ X, kx − yk ≤ r}, which is clopen (closed and open) in X. 2.3. Lemma. A sequence of weak distributions {µLn : n} is generated by some measure µ on B f (X) if and only if for each c > 0 there exists b > 0 such that ||µLn |(B(X, 0, r) ∩ Ln ) − |µLn |(Ln )| ≤ c for each r ≥ b and every n ∈ N and sup |µLn |(Ln ) < ∞. n

6

Sergey V. Ludkovsky

Proof. We consider the non-trivial case, when µLn are non-zero. In the case of µ with values in R we can use the Jordan decomposition µ = µ+ − µ− of the measure µ, where µ+ and µ− are non-negative measures so that µ+ (A) := supB⊂A,B∈E µ(B) and µ− (A) := − infB⊂A,B∈E µ(B) and |µ|(A) = µ+ (A) + µ− (A) for A ∈ E, since X is the Radon space in view of Theorem 1.2 §I.1.3 [DF91]. Recall that the class K X of all countably compact subsets in X is compact, that is for T / there exists a natural number m ∈ N so that each sequence Kn in K X with ∞ n=1 Kn = 0 Tm / A topological space X is called countably compact if from each countable n=1 Kn = 0. open covering of X it is possible to choose a finite sub-covering of X [Eng86]. A measure µ on X is called Radon if it is approximated from below by the class K X , that is for each A ∈ E and every ε > 0 there exists K ∈ K X such that |µ|(A \ K) < ε. Therefore, reduce the proof to the variant, when the measure µ is non-negative and µ(X) = 1, since for 0 < µ(X) < ∞ substitute µ(A) on µ(A)/µ(X) in case of necessity. If a sequence of weak distributions {µLn } is generated by the measure µ, then for 0 < ε < 1 choose r > 0 such that µ(B(X, 0, r)) > 1 − ε, since limr→∞ µ(B(X, 0, r)) = 1. Therefore, µL (B(X, 0, R)∩L) = µ(PL−1 (B(X, 0, R)∩L)) ≥ µ(B(X, 0, R)) ≥ µ(B(X, 0, r)) > 1−ε for each R > r. Vice versa let a sequence of weak distributions satisfying conditions of the lemma be given. On the algebra B0 define a finitely additive mapping µ(A) := µL (A) for A ∈ BL . T Consider a cylindrical set An with a base Bn in Ln so that n An = 0/ and An+1 ⊂ An for each n. For each n choose a closed set Cn such that Cn ⊂ Bn and µLn (Bn \ Cn ) < εn . Then T take Dn := nm=1 {x : PLm x ∈ Cm } ∩ Ln , hence Dn is closed and n

µLn (Bn \ Dn ) ≤

∑

n

µLm (Bm \Cm ) ≤

m=1

∑ εm .

m=1

For sets Jn := PL−1 (Dn ) we get Jn+1 ⊂ Jn and n Jn = n An and µ(An ) ≤ µ(Jn ) + ∑nm=1 εm . n If limn→∞ µ(Jn ) = 0, then limn→∞ µ(An ) = 0, since ∑nm=1 εm can be taken arbitrary small. Therefore, we can consider the variant when each base Bn is closed. Hence each An is weakly closed in X, since the field K is sphericallty complete and the toplogically dual spoace of all continuous K-linear functionals on X separates points in X. Then for each 0 < r < ∞ the closed ball B(X, x, r) is weakly closed and weakly compact, since K is locally T T / then ∞ / Then there compact. Since B(X, 0, R) ∩ ∞ n=1 An = 0, n=1 (B(X, 0, R) ∩ An ) = 0. / since each such set B(X, 0, R) ∩ An exists a natural niumber n so that B(X, 0, R) ∩ An = 0, is weakly compact and An+1 ⊂ An for each n. Thus µ(An ) = µLn (An ) ≤ µLn (Ln ) − µLn (Ln ∩ B(X, 0, R)) ≤ ε for r < R. Since ε is arbitrary then limn→∞ µ(An ) = 0, consequently, µ is σ-additive. 2.4. Definition and notations. A function φ : X → R of the form φ(x) = φS (PS x) is called a cylindrical function if φS is a B f (S)-measurable function on a finite-dimensional over K space S in X. For φS ∈ L1 (S, µ, R) for µ with values in R we can define an integral by a sequence of weak distributions {µSn : n}: T

Z

X

φ(x)µ∗ (dx) :=

Z

Sn

T

φSn (x)µSn (dx),

where L p (S, µ, R) denotes the space of all µ-measurable real-valued functions on S with R finite norm k f k p := [ S | f (x)| p |µ|(dx)]1/p , where 1 ≤ p < ∞.

Real-Valued Measures

7

2.5. Lemma. A subset A ⊂ X = c0 (ω0 , K) is relatively compact if and only if A is bounded and for each c > 0 there exists a finite-dimensional over K subspace L ⊂ X so that A¯ ⊂ Lc := {y ∈ X : d(y, L) := inf{kx − yk : x ∈ L} ≤ c}. Proof. If A is bounded and for each c > 0 there exists Lc with A¯ ⊂ Lc , then there is a sequence {k( j) : j ∈ N} ⊂ Z such that lim j→∞ k( j) = ∞, A¯ ⊂ {x ∈ X : |x j | ≤ p−k( j) , j = 1, 2, . . . } =: S, but X is Lindel¨of, S is sequentially compact, consequently, A¯ is compact (see §3.10.31 [Eng86]). If A¯ is compact, then for each c > 0 there exists a finite number S m such that A¯ ⊂ mj=1 B(X, x j , c), where x j ∈ X. Therefore, A¯ ⊂ Lc for L = spanK (x j : j = 1, . . . , m) := (x = ∑mj=1 b j x j : b j ∈ K). 2.6. Remarks and definitions. When the characteristic of the field K is zero, char(K) = 0, then K as the Qp linear space is isomorphic with Qnp , where n ∈ N := {1, 2, . . . }. The topologically dual space over Qp (that is, of continuous linear functionals f : K → Qp ) is isomorphic with Qnp [HR79]. For x and z ∈ Qnp we denote by (z, x) the following sum ∑nj=1 x j z j , where x = (x j : j = 1, . . . , n), x j ∈ Qp . Each number y ∈ Qp has a decomposition y = ∑l al pl , where min(l : al 6= 0) =: ord p (y) > −∞ (ord(0) := ∞) [NB85], al ∈ (0, 1, . . . , p − 1), we define a symbol {y} p := ∑l 1 and {y} p = 0 for |y| p ≤ 1. For a locally compact field K with a characteristic char(K) = p > 0 let π j (x) := a j for each x = ∑ j a j θ j ∈ K (see Notation). For ξ ∈ K∗ we denote ξ(x) also by (ξ, x). All continuous characters χ : K → C have the form −1 η((ξ,x))

χξ (x) = εz

(1)

for each η((ξ, x)) 6= 0, χξ (x) := 1 for η((ξ, x)) = 0, where ε = 1z is a root of unity, z = pord(η((ξ,x))) , π j : K → R, η(x) := {x} p and ξ ∈ Qnp ∗ = Qnp for char(K) = 0, η(x) := π−1 (x)/p and ξ ∈ K∗ = K for char(K) = p > 0, x ∈ K, (see §25 [HR79]). Each character χ is locally constant, hence χ : K → T is also continuous, where T denotes the discrete group of all roots of 1 (by multiplication). For a measure µ there exists a characteristic functional (that is, called the FourierStieltjes transformation) θ = θµ : C(X, K) → C: θ( f ) :=

Z

X

χe ( f (x))µ(dx),

(2)

where either e = (1, . . . , 1) ∈ Qnp for char(K) = 0, or e = 1 ∈ K∗ for char(K) = p > 0, x ∈ X, f is in the space C(X, K) of continuous functions from X into K, in particular for z = f in the topologically dual (conjugate) space X ∗ over K of all continuous K-linear functionals on X, z : X → K, z ∈ X ∗ , θ(z) =: µˆ (z). It has the following properties: θ(0) = 1 for µ(X) = 1

(3a)

sup |θ( f )| = 1 for probability measures ;

(3b)

and θ( f ) is bounded on C(X, K); f

θ(z) is weakly continuous, that is, (X ∗ , σ(X ∗ , X))-continuous,

(4)

8

Sergey V. Ludkovsky

σ(X ∗ , X) denotes a weak topology on X ∗ , induced by the Banach space X over K. To each x ∈ X there corresponds a continuous linear functional x∗ : X ∗ → K, x∗ (z) := z(x), moreover, θ( f ) is uniformly continuous relative to the norm on Cb (X, K) := { f ∈ C(X, K) : k f k := sup | f (x)|K < ∞}; x∈X

θ(z) is positive definite on X ∗ and on C(X, K)

(5)

for µ with values in [0, ∞). Property (4) follows from Lemma 2.3, boundedness and continuity of χe and the fact that due to the Hahn-Banach theorem there is xz ∈ X with z(xz ) = 1 for z 6= 0 such that z|(X⊖L) = 0 and Z Z θ(z) =

X

χe (PL (x))µ(dx) =

L

χe (y)µL (dy),

where L = Kxz , also due to the Lebesgue theorem 2.4.9 [Fed69] for real measures. Indeed, for each c > 0 there exists a compact subset S ⊂ X such that |µ|(X \ S) < c, each bounded subset A ⊂ X ∗ is uniformly equicontinuous on S (see (9.5.4) and Exer. 9.202 [NB85]), that is, {χe (z(x)) : z ∈ A} is the uniformly equicontinuous family (by x ∈ S). On the other hand, χe ( f (x)) is uniformly equicontinuous on a bounded A ⊂ Cb (X, K) by x ∈ S. Property (5) is accomplished, since N

∑

l, j=1

θ( fl − f j )αl α¯ j =

Z

N

| ∑ α j χe ( f j (x))|2 µ(dx) ≥ 0,

X j=1

particularly, for f j = z j ∈ X, where α¯ j is a complex conjugated number to α j . We call a functional θ finite-dimensionally concentrated, if there exists L ⊂ X, dimK L < ℵ0 , such that θ|(X\L) = µ(X). For each c > 0 and δ > 0 in view of Theorem I.1.2 [DF91] and Lemma 2.5 there exists a finite-dimensional over K subspace L and compact S ⊂ Lδ such that kX \ Skµ < c. Let θL (z) := θ(PL z). This definition is correct, since L ⊂ X, X has the isometrical embedding into X ∗ as the normed space associated with the fixed basis of X, such that functionals z ∈ X separate points in X. If z ∈ L, then |θ(z) − θL (z)| ≤ c × b × q, where b = kXkµ , q is independent of c and b. Each characteristic functional θL (z) is Runiformly continuous by z ∈ L relative to the norm k ∗ k on L, since |θL (z) − θL (y)| ≤ | S′ ∩L [χe (z(x)) − χe (y(x))] R µL (dx)| +| L\S′ [χe (z(x)) − χe (y(x))] µL (dx)|, where the second term does not exceed 2C′ for kL\S′ kµL < c′ for a suitable compact subset S′ ⊂ X and χe (z(x)) is an uniformly equicontinuous by x ∈ S′ family relative to z ∈ B(L, 0, 1). Therefore, θ(z) = lim θn (z) (6) n→∞

for each finite-dimensional over K subspace L, where θn (z) is uniformly equicontinuous and S finite-dimensionally concentrated on Ln ⊂ X, z ∈ X, cl( n Ln ) = X, Ln ⊂ Ln+1 for every n, for each c > 0 there are n and q > 0 such that |θ(z) − θ j (z)| ≤ cbq for z ∈ L j and j > n, q = const > 0 is independent from j, c and b. Let {e j : j ∈ N} be the standard orthonormal in the non-Archimedean sense basis in X, e j = (0, . . . , 0, 1, 0, . . . ) with 1 in j-th place. Using countable additivity of µ, local constantness of χe , considering all z = be j and b ∈ K, we

Real-Valued Measures

9

get that θ(z) on X is non-trivial, whilst µ is a non-zero measure, since due to Lemma 2.3 µ is characterized uniquely by {µLn }. Indeed, for µ with values in R a measure µV on V , dimKV < ℵ0 , this follows from the properties of the Fourier transformation F on spaces of generalized functions and also on L2 (V, µV , C) (see §7 [VVZ94]), where F(g)(z) := lim

Z

r→∞ B(V,0,r)

χe (z(x))g(x)m(dx),

z ∈ V, m is the Haar measure on V with values in R. Therefore, the mapping µ 7→ θµ is injective. 2.7. Proposition. Let X = Kj , j ∈ N, (a). µ and Rν be real probability measures on X, suppose ν is symmetric. Then R ˆ (x)µ(dx) ∈ R and for each 0 < l < 1 is accomplished the following µ ˆ X (x)ν(dx) = X ν inequality: Z µ([x ∈ X : νˆ (x) ≤ l]) ≤ (1 − µˆ (x))ν(dx)/(1 − l). X

(b). For each real probability measure µ on X there exists r > p3 such that for each R > r and t > 0 the following inequality is accomplished: µ([x ∈ X :

kxk ≥ tR]) ≤ c

Z

[1 − µˆ (yξ)]ν(dy), X

where ν(dx) = C × exp(−|x|2 )m(dx), m is the Haar measure on X with values in [0, ∞) so that m(B(X, 0, 1)) = 1, ν(X) = 1, 2 > c = const ≥ 1 is independent from t, c = c(r) is non-increasing whilst r is increasing, C > 0. Proof. R(a). Recall that νR is symmetric, if ν(B) = ν(−B) for eachR B ∈ B f (X). Therefore, X χe (z(x))ν(dx) = X χe (−z(x))ν(dx), that is equivalent to X [χe (z(x)) − χe (−z(x))]ν(dx) = 0 Ror νˆ (z) ∈ R. If 0 < l < 1, R then µ([x ∈ X : νˆ (x) ≤ l]) = µ([x : 1 − νˆ (x) ≥ 1 − l]) ≤ X (1 − νˆ (x))µ(dx)/(1 − l) = X (1 − µˆ (x))ν(dx)/(1 − l) due to the Fubini theorem. (b). Let ν(dx) = γ(x)m(dx), where γ(x) = C × exp(−|x|2 ), C > 0, ν(X) = 1. Then F(γ)(z) =: γˆ (z) ≥ 0, and γˆ (0) = 1 and γ is the continuous positive definite function with γ(z) → 0 whilst |z| → ∞. In view of (a) we deduce that µ([x : kxk ≥ tR]) ≤

Z

[1 − µˆ (yξ)]ν(dy)/(1 − l), X

where |ξ| = 1/t, t > 0, l = l(R). Estimating integrals, we get (b). 2.8. Lemma. Let in the notation of Proposition 2.7 νξ (dx) = γξ (x)m(dx), γξ (x) = C(ξ) exp(−|xξ|2 ), νξ (X) = 1, ξ 6= 0, then a measure νξ is weakly converging to the Dirac measure δ0 with the support in 0 ∈ X for |ξ| → ∞. Proof. We have: C(ξ)−1 = Cq (ξ)−1 =

∑ [plq − p(l−1)q ] exp(−p2l |ξ|2) < ∞,

l∈Z

where the sum by l < 0 does not exceed 1, q = jn, j = dimK X, n = dimQp K. Here K is considered as the Banach space Qnp with the following norm | ∗ | p equivalent to | ∗ |K , for

10

Sergey V. Ludkovsky

x = (x1 , . . . , x j ) ∈ X with xl ∈ K as usually |x| p = max1≤l≤ j |xl | p , for y = (y1 , . . . , yn ) ∈ K with yl ∈ Qp : |y| p := max1≤l≤n |yl |Qp . Further we infer that ! Z pl+s

∑

xl 6=0

−s−1

exp 2πi

∑

xi pi+s

i=l

1

=

pl+s

exp(2πiφ)dφ + β(s),

where s + l < 0, lims→−∞ (β(s)p−s−l ) = 0, therefore, sup[|ˆγ1 (z)|R |z|X : z ∈ X, |z| ≥ p3 ] ≤ 2. Then taking 0 6= ξ ∈ K and carrying out the substitution of variable for continuous and bounded functions f : X → R we get lim

Z

|ξ|→∞ X

f (x)νξ (dx) = f (0).

This means that νξ is weakly converging to δ0 for |ξ| → ∞. 2.9. Theorem.Let µ1 and µ2 be two measures in M(X) such that µˆ 1 ( f ) = µˆ 2 ( f ) for each f ∈ Γ, where X = c0 (α, K), α ≤ ω0 , Γ is a vector subspace in a space of continuous functions f : X → K separating points in X. Then µ1 = µ2 . Proof. Remind that a measure defined on the Borel σ-algebra of a topological space is called a Borel measure. The Borel measure µ in the topological space X is called τsmooth, if for each increasing net {U j : j ∈ J} of open subsets U j in X with a directed set S J the equality µ( j∈J U j ) = lim j∈J µ(U j ) is satisfied. In the set J a relation ≤ directs J if it satisfies conditions (D1 − D3): (D1) if x ≤ y and y ≤ z in J, then x ≤ z; (D2) x ≤ x for each x ∈ J; (D3) for all x, y ∈ J there exists an element z ∈ J so that x ≤ z and y ≤ z. In the family M (X) of all Borel probability measures in the topological Tychonoff (completely regular) space X the weak topology is defined by the base of neighborhoods of R R a measure µ ∈ M (X) of the form U(µ; f1 , . . . , fk ; ε) := {ν ∈ M (X) : | X fi dν − X fi dµ| < ε, i = 1, . . . , k}, where k ∈ N, ε > 0, f1 , . . . , fk are functions from the space Cb (X, C) of all bounded continuous complex-valued functions on X. Proposition 4.5 §I.4[VTC85] states that the set M τ (X) of all τ-smooth probability measures in a topological group X with the convolution operation of measures and the weak topology in M τ (X) is the topological semigroup with the neutral element δ0 . This semigroup M τ (X) is Abelian, if the group X is Abelian. Let at first α < ω0 , then due to continuity of the convolution γξ ∗µ j by ξ, and Proposition 4.5 §I.4[VTC85] and Lemma 2.8 we get µ1 = µ2 , since the family Γ generates B f (X). Now let α = ω0 , A = {x ∈ X : ( f1 (x), . . . , fn (x)) ∈ S}, ν j be an image of a measure µ j for a mapping x 7→ ( f1 (x), . . . , fn (x)), where S ∈ B f (Kn ), f j ∈ X ֒→ X ∗ . Then νˆ 1 (y) = µˆ 1 (y1 f1 + · · · + yn fn ) = µˆ 2 (y1 f1 + · · · + yn fn ) = νˆ 2 (y) for each y = (y1 , . . . , yn ) ∈ Kn , consequently, ν1 = ν2 on E. Let X be a set and Γ be a subset in the family RX of all real-valued mappings from X. The set C f1 ,..., fk ;B := {x ∈ X : ( f1 (x), . . . , fk (x)) ∈ B} with B ∈ B f (Rk ) is called the cylindrical set relative to the pair (X, Γ). The Borel set B is called the base of the cylinder and f1 , . . . , fk are called its generator elements. The set of all cylinders is denoted by C (X, Γ). Remind that a finite Borel measure µ on a Hausdorff topological space X is called Radon, if µ(B) = sup{µ(K) : K ⊂ B, K is compact} for each B ∈ B f (X). If the latter condition is satisfied for the set B = X, then the measure µ is called tight.

Real-Valued Measures

11

The Prohorof’s theorem 3.4 §1.3 [VTC85] states that if X is a completely regular Hausdorff topological space, Γ is some family of real-valued continuous functions from X separating points in X, if also a finitely additive function µ : C (X, Γ) → [0, 1] with µ(X) = 1 is regular and tight, then there exists a unique extension of µ up to a Radon probability measure on X. Further we can use the Prohorov theorem 3.4 §1.3 [VTC85], since compositions of f ∈ Γ with continuous functions g : K → R generate a family of real-valued functions separating points of X. This finishes the proof. 2.10. Proposition. Let µl and µ be measures in M(Xl ) and M(X) respectively, where Xl = c0 (αl , K), αl ≤ ω0 , X = ∏n1 Xl , n ∈ N. Then the condition µˆ (z1 , . . . , zn ) = ∏nl=1 µˆ l (zl ) for each (z1 , . . . , zn ) ∈ X ֒→ X ∗ is equivalent to µ = ∏nl=1 µl .R Proof. Let µ = ∏nl=1 µl , then µˆ (z1 , . . . , zn ) = X χe (∑ zl (xl )) ∏nl=1 µl (dxl ) = n R ∏l=1 Xl χe (zl (xl ))µl (dxl ). The reverse statement follows from Theorem 2.9. 2.11. Proposition. Let X be a Banach space over K; suppose µ, µ1 and µ2 are probability measures on X. Then the following conditions are equivalent: µ is the convolution of two measures µ j , µ = µ1 ∗ µ2 , and µˆ (z) = µˆ 1 (z)ˆµ2 (z) for each z ∈ X. Proof. Let µ = µ1 ∗ µ2 . This means by the definition that µ is the image of the measure µ1 ⊗ µ2 for the mapping (x1 , x2 ) 7→ x1 + x2 , x j ∈ X, consequently, µˆ (z) =

Z 2

X×X

=∏

Z

l=1 X

χe (z(x1 + x2 ))(µ1 ⊗ µ2 )(d(x1 , x2 )) χe (z(xl ))µl (dxl ) = µˆ 1 (z)ˆµ2 (z).

On the other hand, if µˆ 1 µˆ 2 = µˆ , then µˆ = (µ1 ∗ µ2 )∧ and due to Theorem 2.9 above for real measures we have µ = µ1 ∗ µ2 . 2.12. Corollary. Let ν be a probability measure on the Borel σ-algebra B f (X) and µ ∗ ν = µ for each µ, then ν = δ0 . Proof. If z0 ∈ X ֒→ X ∗ and µˆ (z0 ) 6= 0, then from µˆ (z0 )νˆ (z0 ) = µˆ (z0 ) it follows that νˆ 0 (z0 ) = 1. From the property 2.6(6) we get that there exists m ∈ N with µˆ (z) 6= 0 for each z with kzk = p−m , since µˆ (0) = 1. Then νˆ (z + z0 ) = 1, that is, νˆ |(B(X,z0 ,p−m )) = 1. Since µ are arbitrary we get νˆ |X = 1, so we infer that ν = δ0 due to §2.6 and §2.9. 2.13. Corollary. Let X and Y be Banach spaces over K, (a) let µ and ν be probability measures on X and Y respectively, suppose T : X → Y is a continuous linear operator. A measure ν is an image of µ for T if and only if νˆ = µˆ ◦ T ∗ , where T ∗ : Y ∗ → X ∗ is an adjoint operator. (b). A characteristic functional of a real measure µ on B f (X) is real if and only if µ is symmetric. Proof. It follows immediately from §2.6 and §2.9. 2.14. Definition. We say that a real probability measure µ on B f (X) for a Banach space R X over K and 0 < q < ∞ has a weak q-th order if ψq,µ (z) = X |z(x)|q µ(dx) < ∞ for each z ∈ X ∗ . The weakest vector topology in X ∗ relative to which all (ψq,µ : µ) are continuous is denoted by τq := τq (X ∗ , X). 2.15. Theorem. A characteristic functional µˆ of a real probability Radon measure µ on B f (X) is continuous in the topology τq for each q > 0.

12

Sergey V. Ludkovsky Proof. For each c > 0 there exists a compact subset Sc := S ⊂ X such that µ(S) > 1−c/4

and |1 − µˆ (z)| ≤ |

Z

S

(1 − χe (z(x)))µ(dx)| + |

Z

X\S

(1 − χe (z(x)))µ(dx)| ≤ |1 − µˆ c (z)| + c/2,

where µc (A) = (µ(A ∩ S)/µ(S) and A ∈ B f (X). Define the measure µc := µ(B ∩ Sc )/µ(Sc ) for each B ∈ B f (X). Then the measure µc has the compact support, consequently, it has any R strong order p, 0 < p < ∞, that is by the definition X kxk p dµc (x) < ∞. Since τ p1 (X ∗ , X) ⊂ τ p2 (X ∗ , X) for each 0 < p1 < p2 < ∞, then we can consider 0 < p ≤ 1 without loss of generality. In view of the inequality |1 − exp(it)| ≤ 2|t| p for each t ∈ R we deduce that |1 − µˆ c (x∗ )| ≤ 2ψ p,µc (x∗ ) for every x∗ ∈ X ∗ , where i = (−1)1/2 . Thus |1 − µˆ (x∗ )| ≤ 2ψ p,µc (x∗ ) + c/2 for each x∗ ∈ X ∗ . Therefore, if ψ p,µc (x∗ ) < c/4, then |1 − µˆ (x∗ )| < c, consequently, the characteristic functional µˆ is continuous at zero in the topology τ p (X ∗ , X). 2.16. Proposition. For a completely regular space X with zero small inductive dimension ind(X) = 0 the following statements are accomplished: (a). if (µβ ) is a bounded net of measures in M(X) that weakly converges to a measure µ in M(X), then (ˆµβ ( f )) converges to µˆ ( f ) for each continuous f : X → K; if X is separable and metrizable then (ˆµβ ) converges to µˆ uniformly on subsets that are uniformly equicontinuous in C(X, K); (b). if M is a bounded dense family in a ball of the space M(X) for measures in M(X), then a family (ˆµ : µ ∈ M) is equicontinuous on a locally K-convex space C(X, K) in a topology of uniform convergence on compact subsets S ⊂ X. Proof. (a). Functions χe ( f (x)) are continuous and bounded on X, where µˆ ( f ) = R χ X e ( f (x))µ(dx). The Ranga Rao’s proposition 1.3.9[VTC85] states that if a net {µ j } of Borel probability measures in a separable metric space X weakly converges to a Borel probability measure µ inR X and if RΓ ⊂ Cb (X, C) is a bounded pointwise equicontinuous subset, then lim j sup f ∈Γ | X f dµ j − X f dµ| = 0. Then Statement (a) follows from the definition of the weak convergence and Proposition 1.3.9[VTC85], since the linear span spanC {χe ( f (x)) : f ∈ C(X, K)} over the complex field is dense in C(X, C). (b). For each c > 0 there exists a compact subset Sc := S ⊂ X such that |µ|(S) > |µ(X)|− c/4. The set Vc := { f ∈ C(X, K) : | f (x)| < c1/2 ∀x ∈ Sc } is a neighborhood of zero in the topology τc in C(X, K) of the uniform convergence on compact subsets in X. Without loss of generality we can consider µ(X) = 1 renormalizing µ in case of necessity. Therefore, for µ ∈ M and f ∈ C(X, K) with | f (x)|K < c < 1 for x ∈ S we get |µ(X) − Re(ˆµ( f )| = 2| ≤ 2| +2|

Z

Z

1/2

S

1/2

X

1/2

[(χe ( f (x)) − χe (− f (x))/(2i)]2 µ(dx)| 1/2

[(χe ( f (x)) − χe (− f (x))/(2i)]2 µ(dx)| 1/2

X\S

Z

1/2

[(χe ( f (x)) − χe (− f (x))/(2i)]2 µ(dx)| < c.

We have that X is the T1 -space and for each point x and each closed subset S in X with x ∈ /S there is a continuous function h : X → B(K, 0, 1) such that h(x) = 0 and h(S) = {1}. Thus the family {Reˆµ : µ ∈ M(X)} is equicontinuous at zero in C(X, R) in the topology τc .

Real-Valued Measures

13

2.17. Theorem. Let X be a Banach space over K, γ : Γ → C be a continuous positive definite function, (µβ ) be a bounded weakly relatively compact net in the space Mt (X) of Radon norm-bounded measures and there exists limβ µˆ β ( f ) = γ( f ) for each f ∈ Γ and ˜ where Γ ⊂ C(X, K) is a vector subspace uniformly on compact subsets of the completion Γ, separating points in X. Then (µβ ) weakly converges to µ ∈ Mt (X) with µˆ |Γ = γ. Proof. We show that µβ has a unique limit point in the space M t (X) of Radon probability measures in X. Suppose the contrary that there exist two distinct limit points µ1 , µ2 ∈ M t (X). Then there would exist two nets µ1,ρ and µ2,φ weakly converging to µ. That is, limρ µˆ 1,ρ = limφ µˆ 2,φ = limβ µˆ β ( f ) = γ( f ), f ∈ Γ. On the other hand, from the weak convergence it follows that limρ µˆ 1,ρ ( f ) = µˆ 1 ( f ) and limφ µˆ 2,φ ( f ) = µˆ 2 ( f ) for each f ∈ Γ. From the letter equalities we deduce that µˆ 2 = µˆ 1 = γ. But in view of Theorem 2.9 we get µ1 = µ2 . Thus {µβ } has a unique limit point µ ∈ M t (X). From the relative compactness of {µβ } it follows that µβ weakly converges to µ and µˆ ( f ) = γ( f ) for every f ∈ Γ. 2.18. Theorem. (a). A bounded family M of measures in M(Kn ) is weakly relatively compact if and only if a family (ˆµ : µ ∈ M) is equicontinuous on Kn . (b). If (µ j : j ∈ N) is a bounded sequence of measures in Mt (Kn ), γ : Kn → C is a continuous and positive definite function, µˆ j (y) → γ(y) for each y ∈ Kn , then (µ j ) weakly converges to a measure µ with µˆ = γ. (c). A bounded sequence of measures (µ j ) in Mt (Kn ) weakly convereges to a measure µ in Mt (Kn ) if and only if for each y ∈ Kn there exists lim j→∞ µˆ j (y) = µˆ (y). (d). If a bounded net {µβ : β ∈ J} in Mt (Kn ) converges uniformly on each bounded subset in Kn , then (µβ ) converges weakly to a measure µ in Mt (Kn ), where n ∈ N. Proof. (a). The relative compactness of the family M implies due to the Prohorov’s theorem 1.3.6[VTC85] and Proposition 2.16 above that the family {ˆµ : µ ∈ M} is equicontinuous. Vise versa if the family {ˆµ : µ ∈ M} is equicontinuous then due to Proposition 2.7 the family M is dense, consequently, it is weakly relatively compact. R (b). We have the following inequality: limm sup j>m µ j ([x ∈ Kn : |x| ≥ tR]) ≤ 2 Kn (1 − Re(η(ξy)))ν(dy) with |ξ| = 1/t due to §2.7 and §2.8. In view of Theorem 2.17 (µ j ) converges weakly to µ with µˆ = γ. (c). If µk weakly converges to µ, then µˆ k (y) converges to µˆ (y) for each vector y ∈ Kn . The converse statement follows from (b) and the bijective correspondence between measures and characteristic functionals. (d). From the condition it follows that the function γ := lim µˆ β is positive definite. The uniform convergence of µˆ β to χ in some neighborhood of zero implies the continuity of γ in this neighborhood. Due to the positive definiteness of γ this in its turn implies that γ is continuous everywhere. The space Kn is the countable union of bounded subsets, for example, of balls. Therefore, there exists a subsequence {βk : k ∈ N} such that limk µˆ βk (y) = γ(y) for each y ∈ Kn . In view of (b) we have that µβk weakly converges to µ, where µ is a measure with µˆ = γ. Suppose now that µβ does not converge to µ relative to the weak topology. Consider a metric ρ the convergence relative to which is equivalent to the weak convergence of measures. This is possible, since the space Cb (X, C) is separable. In view of the NagataSmirnov’s metrization theorem 4.4.7 [Eng86] a topological space is metrizable if and only if it is regular and has a σ-locally finite base. Recall that a family of subsets is called σ-locally finite if it can be presented as a countable union of locally finite families.

14

Sergey V. Ludkovsky

Then ρ(µβ , µ) does not converge to zero, hence there exists c > 0 so that the set J0 := {β ∈ J : ρ(µβ , µ) ≥ c} is co-final with J. The net {ˆµβ : β ∈ J0 } converges to γ uniformly on each bounded subset. In accordance with the fact proved above there exists the sequence {βk : k} ⊂ J0 so that µβk weakly converges to µ. But this is impossible, since ρ(µβk , µ) ≥ ε. Thus limβ∈J ρ(µβ , µ) = 0 and inevitably µβ weakly converges to µ. 2.19. Corollary. If (ˆµβ ) → 1 uniformly on some neighborhood of 0 in Kn for a bounded net of measures µβ in Mt (Kn ), then (µβ ) converges weakly to δ0 . Proof. Since µˆ β is positive definite for each β, then |1 − µˆ β (2y)| = |1 − µˆ β (y) + µˆ β (y) − µˆ β (2y)| ≤ |1 − µˆ β (y)| + [2(1 − Reˆµβ (y))]1/2 . From this it follows that the convergence of the net µˆ β to 1 is uniform on each bounded subset in Kn . Applying Statement (d) of the preceding theorem we deduce the demonstration of this corollary. 2.20. Definition. A family of probability measures M ⊂ Mt (X) for a Banach space X over K is called planely concentrated if for each c > 0 there exists a K-linear subspace S ⊂ X with dimK S = n < ℵ0 such that inf(µ(Sc ) : µ ∈ M) > 1 − c, where Sc := {x ∈ X : infy∈S ky − xk ≤ c}. The Banach space Mt (X) is supplied with the following norm kµk := |µ|(X). 2.21. Lemma. Let S and X be the same as in §2.20; z1 , . . . , zm ∈ X ∗ be a separating family of points in S. Then a set E := Sc ∩ (x ∈ X : |z j (x)| ≤ r j ; j = 1, . . . , m) is bounded for each c > 0 and r1 , . . . , rm ∈ (0, ∞). Proof. A space S is isomorphic with Kn , consequently, p(x) = max(|z j | : j = 1, . . . , m) is the norm in S equivalent to the initial norm. Suppose that in E there exists a sequence {xk } so that limk kxk k = ∞. Then for some 1 ≤ j ≤ n we would have limk |z j (xk )| = ∞ contradicting xk ∈ E for each k ∈ N. Thus E is contained in the ball B(X, 0, R) for some 0 < R < ∞. 2.22. Theorem. Let X be a Banach space over K with a family Γ ⊂ X ∗ separating points in a family of probability measures M ⊂ Mt (X). Then M is weakly relatively compact if and only if a family {µz : µ ∈ M} is weakly relatively compact for each z ∈ Γ and M is planely concentrated, where µz is an image measure on K of a measure µ induced by z. Proof. Remind the following. Suppose that X is a metric space with a metric ρ in it. On the space M (X) introduce the Prohorov’s metric d(µ, ν) := inf{b > 0 : µ(A) ≤ ν(Ab ) + b ∀A ∈ B f (X)}, where Ab := {x ∈ X : ρ(x, A) < b}, b > 0. The Prohorov’s criteria states that if a net {µ j : j ∈ J} of probability measures in the metric space X converges in the metric d to a probability measure µ, then µ j weakly converges to µ. If µ ∈ M τ (X), then from the weak convergence of µ j to µ it follows that lim j d(µ j , µ) = 0. Thus the weak topology in M τ (X) is metrizable. The necessity follows from Lemmas 2.5, 2.21 and the Prohorov’s criteria. Prove the sufficiency. For each c > 0 and n ∈ N find a finite-dimensional subspace c2−n−1 ) is Sn,c ⊂ X so that for each µ ∈ M we have µ(Yn,c ) > 1 − c2−n−1 , where Yn,c := cl(Sn,c the closure of the c2−n−1 open neighborhood of the set Sn,c . Consider now a finite set of functionals x1∗ , . . . , xk∗n ∈ Γ separating points in Sn,c . Choose real numbers r1 , . . . , rkn ∈ (0, ∞) satisfying the condition: infµ∈M µx∗j (B(K, 0, r j )) > 1 − c2−n−1 /kn for each j = 1, . . . , kn , where µx∗j (A) := µ({x ∈ X : x∗j (x) ∈ A}) for each Borel subset A in the field K.. T ∗ In accordance with Lemma 2.21 the set K := ∞ n=1 (Yn,c ∩ {x : |x j (x)| < r j ∀ j = −n−1

c2 1, . . . , kn }) is bounded. The set K is closed and K ⊂ Sn,c

for each c > 0 and n ∈ N.

Real-Valued Measures

15

In view of Lemma 2.5 the set K is compact. Moreover, µ(X \ K) ≤

∞

kn

n=1

j=1

∑ (µ(X \Yn,c ) + ∑ µx∗j (K \ B([K, 0, r j )) ≤ c.

Applying the Prohorov’s criteria we deduce that the family M is weakly relatively compact. 2.23. Theorem. For X and Γ the same as in Theorem 2.22 a sequence {µ j : j ∈ N} ⊂ Mt (X) is weakly convergent to µ ∈ Mt (X) if and only if two conditions are satisfied (a) for each z ∈ Γ there exists lim j→∞ µˆ j (z) = µˆ (z) and (b) a family {µ j } is planely concentrated. Proof. The necessity of these two conditions follows from Theorem 22. Now prove the sufficiency. In view of Theorem 2.18 and Condition (a) we have that (µk )x∗ weakly converges in K for each x∗ ∈ Γ. Due to Theorem 22 the sequence {µk : k} is weakly relatively compact. Applying Theorem 17 we deduce the statement of this theorem. 2.24. Proposition. Let X be a completely regular space with the zero small inductive dimension ind(X) = 0, Γ ⊂ C(X, K) be a vector subspace separating points in X, (µn : n ∈ N) ⊂ Mt (X), µ ∈ Mt (X), limn→∞ µˆ n ( f ) = µˆ ( f ) for each f ∈ Γ. Then (µn ) is weakly convergent to µ relative to the weakest topology σ(X, Γ) in X relative to which all f ∈ Γ are continuous. Proof. Recall the A.D. Alexandroff’s theorem. Let {µ j : j ∈ J} be a net of measures in M (X), µ ∈ M τ (X). If X is a metric space, then the τ-smoothness of µ is not necessary. Then the following conditions are equivalent: (a) µ j weakly converges to µ; (b) lim j µ j (F) ≤ µ(F) for each closed subset F in X; (c) lim j µ j (U) ≥ µ(U) for each open U in X; (d) lim j µ j (B) = µ(B) for each B ∈ B f (X) so that µ(Fr(B)) = 0. In view of A.D. Alexandroff’s theorem it is sufficient to show that lim j µ j (U) ≥ µ(U) for each σ(X, Γ) open U in X. At first show this relation for each open U ∈ C (X, Γ). Consider an arbitrary cylinder U0 := {x ∈ X : ( f1 (x), . . . , fn (x)) ∈ Vn }, where f1 , . . . , fn ∈ Γ, Vn is open in Kn . Under the mapping x 7→ ( f1 (x), . . . , fn (x)) we get the images ν, νn in Kn of measures µ, µn , where n ∈ N. For any y = (y1 , . . . , yn ) ∈ Kn we have ! ! lim νˆ n (y) = lim µˆ n

n

n

∑ yk fk

k=1

n

= µˆ

∑ yk fk

= νˆ (y).

k=1

In view of Theorem 18 νn weakly converges to ν, consequently, lim j ν j (Vn ) ≥ ν(Vn ) and inevitably lim j µ j (U0 ) ≥ µ(U0 ). Let now U be an arbitrary open subset and U0 ⊂ U. Then lim j µ j (U) ≥ lim j µ j (U0 ) ≥ µ(U0 ). By the definition of the topology σ(X, Γ) each σ(X, Γ)-open subset U in X is a union of some family of open cylindrical subsets. Denote by U the family of all open cylinders U0 ⊂ U. The family U is ordered by inclusion and the measure µ is τ-smooth, hence µ(U) = supU0 ∈U µ(U0 ), consequently, lim j µ j (U) ≥ µ(U). 2.25. Let (X, U) = ∏λ (Xλ , Uλ ) be a product of measurable completely regular Radon spaces (Xλ , Uλ ) = (Xλ , Uλ , Kλ ), where Kλ are compact classes approximating from below

16

Sergey V. Ludkovsky

each measure µλ on (Xλ , Uλ ), that is, for each c > 0 and elements A of an algebra Uλ there is S ∈ Kλ , S ⊂ A with kA \ Skµλ < c. 2.26. Definition. Let X be a Banach space over K, then a mapping f : X → C is called pseudo-continuous, if its restriction f |L is uniformly continuous for each subspace L ⊂ X with the finite dimension dimK L < ℵ0 over the field K. ˆ Γ) Let Γ be a family of mappings f : Y → K of a set Y into a field K. We denote by C(Y, the minimal σ-algebra (that is called cylindrical) generated by an algebra C(Y, Γ) of subsets of the form C f1 ,..., fn ;E := {x ∈ X : ( f1 (x), . . . , fn (x)) ∈ S}, where S ∈ B f (Kn ), f j ∈ Γ. We supply Y with a topology τ(Y ) which is generated by a base (C f1 ,..., fn ;E : f j ∈ Γ, E is open in Kn ). 2.27. Theorem. Non-Archimedean analog of the Bochner-Kolmogorov theorem. Let X be a Banach space over K, X a be its algebraically dual K-linear space (that is, of all linear mappings f : X → K not necessarily continuous). A mapping θ : X a → C is a characteristic functional of a probability measure µ with values in R and is defined ˆ a , X) if and only if θ satisfies conditions 2.6(3, 5) for (X a , τ(X a )) and is pseudoon C(X continuous on X a . Proof. (I). For dimK X = card(α) < ℵ0 a space X a is isomorphic with Kα , hence the statement of theorem for a measure µ follows from Theorems 2.9 and 2.18 above, since θ(0) = 1 and |θ(z)| ≤ 1 for each z. (II). We consider now the case of µ with values in R and α < ω0 . In §2.6 (see also §2.16-18,24) it was proved that θ = µˆ has the desired properties for real probability measures µ. On the other hand, there is θ which satisfies the conditions of the theorem. Let θξ (y) = θ(y)hξ (y), where hξ (y) = F[C(ξ) exp(−kxξk2 )](y) (that is, the Fourier transform by x), νξ (Kα ) = 1, νξ (dx) = C(ξ) exp(−kxξk2 )m(dx) (see Lemma 2.8), ξ 6= 0. Then θξ (y) is positive definite and is uniformly continuous as a product of two such functions. Moreover, θξ (y) ∈ L1 (Kα , m, C). For ξ 6= 0 a function fξ (x) = R 2 Kα θξ (y)χe (x(y))m(dy) is bounded and continuous, the function exp(−kxξk ) =: s(x) is positive definite. Since νξ is symmetric and weaklyR converges to δ0 , hence there exists r > 0 such that for each |ξ| > r we have γˆ ξ (y) = Kα C(ξ) exp(−k xξk2p )χe (y(x))m(dx) R = [χe ((y(x)) + χe (−y(x))]2−1 exp(−kxξk2p )C( ξ)m(dx)/2 > 1 − 1/R for |y| ≤ R, consequently, γˆ ξ (y) = ζˆ 2ξ (y) for |y| ≤ R, where ζˆ ξ is positive definite uniformly continuous and has a uniformly continuous extension on Kα . Therefore, for each c > 0 there exists r > 0 such that kνξ − κξ ∗ κξ k < c for each |ξ| > r, where κξ (dx) = ζξ (x)m(dx) is a σ-additive non-negative measure. Recall the following proposition about positive definite functions (see also IV.1.3 [VTC85]). Let (Ω, B ) be a measurable space, ν be a σ-finite measure on B and let f : Ω × Ω → C be a measurable and ν × ν-integrable positive definite function, then R Ω×Ω f d(ν × ν) ≥ 0. From this proposition we also use the following corollary. Let (Ω, B ) be a measurable group, let also ν be a symmetricRprobability measure on B and let g : Ω → C be a measurable positive definite function, then Ω gd(ν ∗ ν) ≥ 0. R Hence due to this corollary there exists r > 0R such that Kα θξ (y)χe (−x(y))ν j (dy) ≥ 0 for each | j| > r, consequently, fξ (x) = lim| j|→∞ Kα θξ (y) χe (−x(y))ν j (dy) ≥ 0. From

Real-Valued Measures

17

the equality F[F(γξ )(−y)](x) = γξ (x) and the Fubini theorem it follows that Z

fξ χe (y(x))h j (x)m(dx) =

Z

θξ (u + y)ν j (du).

For y = 0 we get lim

|ξ|→∞

Z

fξ (x)m(dx) =

Z

f (x)m(dx) = lim lim

and lim lim | |ξ|→∞ | j|→∞

|ξ|→∞ | j|→∞

Z

Kα

Z

fξ (x)h j (x)m(dx)

θξ (u)ν j (du)| ≤ 1.

From Lemma 2.8 it follows that fˆ(y) = θ(y), since by Theorem 2.18 θ = lim|ξ|→∞ θξ is a characteristic function of a probability measure on B f (Kα ), where f (x) = R Kα θ(y)χe (−x(y))m(dy). (III). Now let α = ω0 . It remains to show that the conditions imposed on θ are sufficient, because their necessity follows from the modification of 2.6 (since X has an algebraic embedding into X a ). The space X a is isomorphic with KΛ which is the space of all Kvalued functions defined on the Hamel basis Λ in X. The Hamel basis exists due to the Kuratowski-Zorn lemma (that is, each finite system of vectors in Λ is linearly independent over K, each vector in X is a finite linear combination over K of elements from Λ). Let J be a family of all non-void subsets in Λ. For each A ∈ J there exists a functional θA : KA → C such that θA (t) = θ(∑y∈A t(y)y) for t ∈ KA . From the conditions imposed on θ it follows that θA (0) = 1, θA is uniformly continuous and bounded on KA , moreover, it is positive definite (or due to 2.6(6) for each c > 0 there are n and q > 0 such that for each j > n and z ∈ KA the following inequality is satisfied: (i) |θA (z) − θ j (z)| ≤ cbq, moreover, L j ⊃ KA , q is independent on j, c and b. From (I,II) it follows that on B f (KA ) there exists a probability measure µA such that µˆ A = θA . The family of measures {µA : A ∈ J} is consistent and bounded, since µA = µE ◦ (PEA )−1 , if A ⊂ E, where PEA : KE → KA are the natural projectors. Indeed, in the case of measures with values in R each µA is the probability measure. Remind the Kolmogorov’s theorem (see also Theorem 1.1.4 [DF91]). Suppose that (X, U ) = ∏ j∈Λ (X j , U j ) be the product of measurable Radon spaces (X j , U j ) = (X j , U j , K j ), then each bounded quasi-measure on (X, U ) is a measure. In view of the Kolmogorov’s theorem on a cylindrical σ-algebra of the space KΛ there exists the unique measure µ such that µA = µ ◦ (PA )−1 for each A ∈ J, where PA : KΛ → KA ˆ a , X). For µ on are the natural projectors. From X a = KΛ it follows that µ is defined on C(X ˆ a , X) there exists its extension on A f (X, µ) (see §2.1). C(X 2.28. Definition. [Sch89] A continuous linear operator T : X → Y for Banach spaces X and Y over K is called compact, if T (B(X, 0, 1)) =: S is a compactoid, that is, for each neighbourhood U ∋ 0 in Y there exists a finite subset A ⊂ Y such that S ⊂ U + co(A), where co(A) is the least K-absolutely convex subset in V containing A (that is, for each a and b ∈ K with |a| ≤ 1, |b| ≤ 1 and for each x, y ∈ V the following inclusion ax + by ∈ V is accomplished).

18

Sergey V. Ludkovsky

2.29. Let B+ be a subset of non-negative functions which are B f (X)-measurable and ˆ + we denote a family of let C+ be its subset of non-negative cylindrical functions. By B ˆ + let functions f ∈ B+ such that f (x) = limn gn (x), gn ∈ C+ , gn ≥ f . For f ∈ B Z

f (x)µ∗ (dx) :=

X

inf

Z

g≥ f , g∈C+ X

g(x)µ∗ (dx).

2.30. Lemma. A sequence of weak distributions (µLn ) of probability Radon measures is generated by a real probability measure µ on B f (X) of a Banach space X over K if and only if there exists Z (i) lim

|ξ|→∞ X

where

R

X

Gξ (x)µ∗ (dx) = 1,

Gξ (x)µ∗ (dx) := Sξ ({µLn : n}) and Sξ ({µLn }) := lim

Z

n→∞ Ln

Fn (γξ,n )(x) µLn (dx),

γξ,n (y) :=

m(n)

∏ γξ (yl ), l=1

Fn is a Fourier transformation by (y1 , . . . , yn ), y = (y j : j ∈ N), y j ∈ K, γξ (yl ) are the same S as in Lemma 2.8 for K1 ; here m(n) = dimK Ln < ℵ0 , cl( n Ln ) = X = c0 (ω0 , K). Proof. If a sequence of weak distributions is generated by a measure µ, then in view of Conditions 2.6(3-6), Lemmas 2.3, 2.5, 2.8, Propositions 2.10 and 2.16, Corollary 2.13, the Lebesgue convergence theorem and the Fubini theorem, also from the proof of Theorem 2.27 and the Radon property of µ it follows that there exists r > 0 such that Z

X

Gξ (x)µ∗ (dx) =

Z

X

Gξ (x)µ(dx) = lim

Z

n→∞ Ln

γξ,n (y)ˆµLn (y)mLn (dy), R

since lim j→∞ x j = 0 for each x = (x j : j) ∈ X. In addition, lim|ξ|→∞ Sξ ({µLn }) = X µ(dx) = 1. Indeed, for each c > 0 and d > 0 there exists a compact Vc ⊂ X with kµ|(X\Vc ) k < c and there exists n0 with Vc ⊂ Lnd for each n > n0 . Therefore, choosing suitable sequences of R c(n), d(n), Vc(n) and L jn we get that [ Ln γξ,n (y)ˆµLn (y)mLn (dy) : n ∈ N] is a Cauchy sequence, m(n)b

, b = dimQp K, where mLn is the real Haar measure on Ln , the latter is considered as Qp m(B(Ln , 0, 1)) = 1. Here we use Gξ (x) for a formal expression of the limit Sξ as the integral. Then Gξ (x) (mod µ) is defined evidently as a function for µ or {µLn : n} with a compact support, also for µ with a support in a finite-dimensional subspace L over K in X. By the definition supp (µLn : n) is compact, if there is a compact V ⊂ X with supp(µLn ) ⊂ PLn V for each n. That is, Condition (i) is necessary. On the other hand, if (i) is satisfied, then for each c > 0 there exists r > 0 such that R | X Gξ (x)µ∗ (dx) − 1| < c/2, when |ξ| > r, consequently, there exists n0 such that for each n > n0 the following inequality is satisfied: Z 1 − Fn (γξ,n )(x)µ∗ (dx) ≤ |kµ|(L ∩B(X,0,R)) k − 1| n X

+ sup |Fn (γξ,n )(x)|kµLn |(Ln \B(X,0,R)) k. |x|>R

Therefore, from limR→∞ sup|x|>R |Fn (γξ,n )(x)| = 0 and from Lemma 2.3 the statement of Lemma 2.30 follows.

Real-Valued Measures

19

2.31. Notes and definitions. Suppose X is a locally convex space over a locally compact field K with non-trivial non-Archimedean normalization and X ∗ is a topologically dual space. The minimum σ-algebra with respect to which the following family {v∗ : v∗ ∈ X ∗ } is measurable is called a σ-algebra of cylindrical sets. Then X is called a RS-space if on X ∗ there exists a topology τ such that the continuity of each positive definite function f : X ∗ → C is necessary and sufficient for f to be a characteristic functional of a non-negative measure. Such topology is called the R-Sazonov type topology. The class of RS-spaces contains all separable locally convex spaces over K. For example, l ∞ (α, K) = c0 (α, K)∗ , where α is an ordinal [Roo78]. In particular we also write c0 (K) := c0 (ω0 , K) and l ∞ (K) := l ∞ (ω0 , K), where ω0 is the first countable ordinal. Let nK (l ∞ , c0 ) denotes the weakest topology on l ∞ for which all functionals px (y) := supn |xn yn | are continuous, where x = ∑n xn en ∈ c0 and y = ∑n yn e∗n ∈ l ∞ , en is the standard base in c0 . Such topology nK (l ∞ , c0 ) is called the normal topology. The induced topology on c0 is denoted by nK (c0 , c0 ). 2.32. Theorem. Let f : l ∞ (K) → C be a functional such that (i) f is positive definite, (ii) f is continuous in the normal topology nK (l ∞ , c0 ), then f is the characteristic functional of a probability measure on c0 (K). Proof. The case of the topological vector space X over K with char(K) > 0 and a real-valued measure µ can be proved analogously to the proofs in Chapter II for Ks -valued measures as well as for char(K) = 0 due to §2.6 and §§2.25-2.30 (see also [Mad85]). 2.33. Theorem. Let µ be a probability measure on c0 (K), then µˆ is continuous in the normal topology nK (l ∞ , c0 ) on l ∞ . Proof. In view of Lemma 2.3 for each ε > 0 there exists S(ε) ∈ c0 such that kµ|L(0,S(ε)) k ≥ 1 − ε, where L(y, z) := {x ∈ c0 : |xn − yn | ≤ |zn |, for each n ∈ N}. Therefore, |1 − µˆ (x)| ≤ ε + k2πη(ξx)kC0 (L(0,S(ε))) kµ|L(0,S(ε)) k, hence there exists a constant C > 0 such that |1 − µˆ | ≤ ε +CpS(ε) (x). 2.34. Corollary. The normal topology nK (l ∞ , c0 ) is the R-Sazonov type topology on l ∞ (K). 2.35. Theorem. Non-Archimedean analog of the Minlos-Sazonov theorem. For a separable Banach space X over K the following two conditions are equivalent: (I) θ : X → T satisfies conditions 2.6(3, 4, 5) and for each c > 0 there exists a compact operator Sc : X → X such that |Re(θ(y) − θ(x))| < c for |˜z(Sc z)| < 1; (II) θ is a characteristic functional of a probability Radon measure µ on E, where z˜ is an element z ∈ X ֒→ X ∗ considered as an element of X ∗ under the natural embedding associated with the standard base of c0 (ω0 , K), z = x − y, x and y are arbitrary elements of X. Proof. (II → I). For a positive definite function θ generated by a probability measure µ in view of the inequality |θ(y) − θ(x)|2 ≤ 2θ(0)(θ(0) − Re(θ(y − x)) (see also Propositions

20

Sergey V. Ludkovsky

IV.1.1(c)[VTC85]) and using the normalization of a measure µ by 1 we consider the case y = 0. For each r > 0 we have: |Re(θ(0) − θ(x))| = ≤

Z

B(X,0,r)

Z

X

(1 − [χe (x(u)) + χe (−x(u))]/2)µ(du)

1/2 1/2 2[χe (x(u)) − χe (−x(u))/(2i)]2 µ(du) + 2

≤ 2π2

Z

B(X,0,r)

η(x(u))2 µ(dx) + 2µ([x :

Z

µ(du) X\B(X,0,r)

kxk > r]).

In view of the Radon property of the space X and Lemma 2.5 for each b > 0 and δ > 0 there are a finite-dimensional over K subspace L in X and a compact subset W ⊂ X such that W ⊂ Lδ , kµ|(X\W ) k < b, hence kµ|(X\Lδ ) k < b. We consider the following expression: J( j, l) := 2π2

Z

B(X,0,r)

η(e j (u))η(el (u))µ(du),

where (e j ) is the orthonormal basis in X which contains the orthonormal basis of L = Kn , n = dimK L. Then we choose sequences b j = p− j and 0 < δ j < b j , subspaces L j and r = r j such that b j r j < 1, W j ⊂ B(X, 0, r j ), 0 < r j < r j+1 < ∞ for each j ∈ N and the orthonormal basis (e j ) corresponding to the sequence L j ⊂ L j+1 ⊂ · · · ⊂ X. We get, due to finiteness of n j := dimK L j , that lim j+l→∞ J( j, l) = 0, since kµ|{x:kxk>r j } k < b j , η(x(u)) = 0 for x ∈ X ⊖L j with kxk < b j , u ∈ B(X, 0, r j ). Then we define g j,l := min{d : d ∈ ΓK and d ≥ |J( j, l)|}, evidently, g j,l ≤ p|J( j, l)| and there are ξ j,l ∈ K with |ξ j,l |K = g j,l . Consequently, the family (ξ j,l ) determines a compact operator S : X → X with e˜ j (Sel ) = ξ j,l t due to Theorem 1.2[Sch89], where t = const ∈ K, t 6= 0. Therefore, |Re(θ(0) − θ(z))| < c/2 + |˜z(Sz)| < c |θ(0) − θ(z)| < c/2 + |˜z(Sz)| < c. We choose r such that kµ|(X\B(X,0,r)) k < c/2 with S corresponding to (r j : j), where r1 = r, L1 = L, then we take t ∈ K with |t|c = 2. (I → II). Without restriction of generality we may take θ(0) = 1 after renormalization of non-trivial θ. In view of Theorem 2.32 as in §2.6 we construct using θ(z) a consistent family of finite-dimensional distributions {µLn }. Let mLn be a real Haar measure on Ln which is considered as Qap with a = dimK Ln dimQp K, m(B(Ln , 0, 1)) = 1. In view of Proposition 2.7 and Lemmas 2.8, 2.30: Z

Ln

Gξ (x)µLn (dx) =

Z

Ln

γξ,n (z)θ(z)mLn (dz),

consequently, 1−

Z

Ln

Fn (γξ,n )(x)µLn (dx) =

Z

γξ,n (z)(1 − θ(z))mLn (dz) =: In (ξ).

There exists an orthonormal basis in X in which Sc can be reduced to the following form Sc = SCSˆc E (see Appendix), where Sˆc = diag(s j : j ∈ N) in the orthonormal basis ( f j : j) in X and S transposes a finite number of vectors in the orthonormal basis. That is, |˜z(Sˆc z)| = max j |s j | × |z j |2 . In the orthonormal basis (e j : j) adopted to (Ln : n) we have |˜z(Sc z)| = max j,l∈N (|s j,l | × |z j | × |zl |), kSc k = max j,l |s j,l |, where Sc = (s j,l : j, l ∈ N) in the

Real-Valued Measures

21

orthonormal basis (e j ), r = const > 0. In addition, p−1 |x|K ≤ |x| p ≤ p|x|K for each x ∈ K. If Sc is a compact operator such that |Re(θ(y) − θ(x))| < c for |˜z(Sc z)| < 1, z = x − y, then |Re(1 − θ(x))| < c + 2|x(S ˜ c x)| and In (ξ) ≤

Z

γξ,n (z)[c + 2|˜z(Sc z)|K ]mLn (dz) ≤ c + bkSc k/|ξ|2 ,

b = const is independent from n, ξ and Sc , b := p × sup |ξ|

2

|ξ|>r

Z

Ln

γξ,n (z)|z|2p mLn (dz) < ∞.

Due to the formula of changing variables in integrals (A.7[Sch84]) the following equality is valid: Jn (ξ) = In (ξ)Jn (1)/[In (1)|ξ|2 ] for |ξ| 6= 0, where Jn (ξ) = Therefore, 1−

Z

X

Z

Ln

γξ,n (z)|z|2p mLn (dz).

Gξ (x)µ∗ (dx) ≤ c + bkSc k/|ξ|2 .

Then taking the limit with |ξ| → ∞ and then with c → +0 with the help of Lemma 2.30 we get the statement (I → II). 2.36. Definition. Let on a completely regular space X with the small inductive dimension ind(X) = 0 two non-zero real-valued measures µ and ν be given. Then ν is called absolutely continuous relative to µ if ν(A) = 0 for each A ∈ B f (X) with µ(A) = 0 and it is denoted ν ≪ µ. Measures ν and µ are singular to each other if there is F ∈ B f (X) with |µ|(X \ F) = 0 and |ν|(F) = 0 and it is denoted ν ⊥ µ. If ν ≪ µ and µ ≪ ν then they are called equivalent, ν ∼ µ. 2.37. For a Banach space X over a non-Archimedean infinite locally compact field K and σ-algebra B ⊃ B f (X) with a real probability measure µ and a σ-subalgebra B0 ⊂ B a function µ¯ (A|x) satisfying three conditions: (a) µ¯ (A|x) is B0 -measurable by x for each A ∈ B; R (b) µ(A ∩ A0 ) = A0 µ¯ (A|x)µ(dx) =: µA (A0 ) for each A0 ∈ B0 ; (c) µ¯ (A|x) is a measure by A ∈ B for almost all x relative to a measure µ, then it is called a conditional measure corresponding to µ relative to a σ-algebra B0 . Then we define the conditional measure µ(A|x) by Formula (b) and we then redefine it on the set of measure zero relative to µ for each A. Let θ(z|x) = θ1 (z|x) + iθ2 (z|x) with Z Z

A0

A0

[[χe (˜z(x)) + χe (−˜z(x))]/2]µ(dx) =

Z

[χe (˜z(x)) − χe (−˜z(x))]/(2i)]µ(dx) =

A0

Z

θ1 (z|x)µ(dx);

A0

θ2 (z|x)µ(dx),

where θ j are measurable functions, z ∈ X, z˜ ∈ X ′ corresponds to z under the natural embedding of X into X ′ = X ∗ , where X ∗ is the topologically dual space of X, x ∈ X, A0 ∈ B0 . Then we choose sets An,k with of diameters diam(An,k ) < p−n , using §§2.7, 2.8, 2.30 and

22

Sergey V. Ludkovsky

the proof of §2.27, using the function Gξ (x), defining sets Bn , G, G1 , we get for each c > 0 that there exists r > 0 such that Re(1 − θ(z|x)) ≤

Z

X

Gξ (y)(1 − [[χe (˜z(y)) + χe (−˜z(y))]/2]µ(dy|x) + 2 ≤

Z

Z

X

(1 − Gξ (y))µ(dy|x)

(2πη2 (˜z(y))Gξ (y)µ(dy|x)/2 + c/2

for each |ξ| > r and x ∈ G1 ∩ G, since from kzk × kyk ≤ 1 it follows that (1 − χe (˜z(y)) = 0 and |1 − exp(it)| ≤ |t| for tR ∈ R, i = (−1)1/2 . For C(z1 ,Rz2 )(x) := X η(˜z1 (y))η(˜z2 (y))Gξ (y)µ(dy|x) we have C(0, 0) = 0 and Since θ(z|x) = C(z, z)(x) = X\B(X,0,r) η2 (˜z(y))Gξ (y)µ(dy|x) for kzk = 1/r > 0. R χ (˜ z (y))µ(dy|x), considering projectors P : X → L onto finite-dimensional subspaces Ln n X e over K, L ⊂ X, we get θL (z|x) corresponding to µL (dx) and µL (dy|x). At the same time θL satisfies Conditions 2.6(3,4,5), consequently, µL (A|x) are σ-additive by A for fixed x, since θ(z|x) are continuous by z for x ∈ G ∩ G1 . Remind the Prohorov’s theorem. Let T = (Tk , pk, j ) be a projective system of topological spaces, where k, j ∈ J, let also T be a topological space and (p j : j ∈ J) be a cogerent pk = pk, j ◦ p j defining family of continuous mappings p j : T → T j , pk, j : T j → Tk for each k ≤ j ∈ J. Suppose that (µ j : j ∈ J) is a projective system of measures on T , µk = pk, j (µ j ) is a bounded measure on Tk for each k ≤ j ∈ J. Then a bounded measure µ on T with p j (µ) = µ j for each j ∈ J exists if and only if the following condition is satisfied: (P) for each c > 0 there exists a compact subset K ⊂ T so that µ∗j (T j \ p j (K)) ≤ c for each j ∈ J. If this condition (P) is satisfied, then µ is unique and µ∗ (K) = inf j µ∗j (p j (K)) for each compact subset K ⊂ T (see also §IX.4.2.1 in [Bou63-69]). From the Prohorov’s theorem, Lemma 2.5, x ∈ G ∩ G1 and C(e j , el )(x) =

Z

L j,l \B(L j,l ,0,1)

it follows that lim

Z

j+l→∞ L j,l

= lim

Z

j+l→∞ L j,l

ψ(x)

Z

L j,l

η(yl )η(y j )F2 (γξ,2 )(y j , yl )µL j,l (dy j dyl |x)

ψ(x)C(e j , el )(x)µL j,l (dx)

η(y j )η(yl )χe (˜z(y))µL j,l (dy|x)µL j,l (dx) = 0

for each bounded B0 -measurable function ψ(x), whence limn→∞ µ([x : |C(e j , el )(x)| > c for some j + l > n]) = 0 for each c > 0, where L j,l = Ke j ⊕ Kel , y j ∈ K. Let G2 := [x : for each c > 0 there exists n > 0 with |C(e j , el )(x)| < c for each j + l > S T S n] ⊂ X \ ( c>0 ∞ n=1 j+l>n [x : |C(e j , el )(x)| > c]), then µ(G2 ) = 1. Therefore, for each x ∈ G ∩ G1 ∩ G2 there exists a compact operator D : X → X such that |C(z1 , z2 )(x)| ≤ |˜z2 (Dz1 )| for each z1 , z2 ∈ X. In view of Theorem 2.35 and the equality Z

[ A0

Z

X

χe (z(y))¯µ(dy|x)]µ(dx) =

Z

A0

χe (z(y))¯µA0 (dy)

we get that Z

X

χe (z(y))¯µA0 (dy) =

Z

X

θ(z, x)µ(dx) =

Z

A0

χe (z(y))µ(dy),

Real-Valued Measures

23

where µ¯ A0 (B) = µ(A0 |B) is the measure on B f (X), so that µ¯ A0 (B) = µ(A0 ∩ B). That is µ(A0 ∩ B) =

Z

µ¯ (B|x)µ(dx),

(1)

A0

hence µ¯ (B|x) is the conditional measure of µ relative to the σ-algebra B0 . Thus the existence of the conditional measure is demonstrated. There is the following important particular case, when the σ-algebra B0 is generated by some finite or infinite family of functions {φ j : j ∈ J}, that is is the minimal σ-algebra relative to which all these functions are measurable. The relation (1) is equivalent with the statement: for each bounded B0 -measurable function ψ(x) and B f (X)-measurable function ψ(x), R for which X |φ(x)|µ(dx) < ∞ the equality Z

X

ψ(x)φ(x)µ(dx) =

Z

X

ψ(x)

Z

X

φ(y)¯µ(dy|x)µ(dx)

(2)

is satisfied. Relation (2) implies that Z

g(y)φ(y)¯µ(dy|x) = g(x) X

Z

X

φ(y)¯µ(dy|x) (mod µ)

(3)

for each bounded B0 -measurable function g(x) and a function φ(x), for which X |φ(x)|µ(dx) < ∞. 2.38. Definition. Martingales. Let (X, B , µ) be a measure space, where µ is a nonnegative measure on a measurable space (X, B ), B is a σ-algebra on a set X. A sequence of measurable real-valued functions {φn (x) : n ∈ N} on (X, B , µ) is called a martingale, if for each n: Z |φn (x)|µ(dx) < ∞, (1) R

X

while for each U n measurable bounded non-negative function ψ(x) ≥ 0, the relation Z

X

φn+1 (x)ψ(x)µ(dx) =

Z

X

φn (x)ψ(x)µ(dx)

(2)

is satisfied, where U n denotes the minimal σ-algebra relative to which functions φ1 , . . . , φn are measurable. If instead of the equality there is the inequality ≥ or ≤, then such sequence is called the sub-martingale or super-martingale respectively. 2.39. Lemma. For each sub-martingale (super-martingale) on a Banach space X over K there exists a sequence of functions gn (x) satisfying the following conditions (a − c): (a) gn (x) are U n -measurable; (b) the sequence φn − gn is the martingale; (b) gn (x) increases with n (decreases with n correspondingly) µ-almost everywhere. Proof. Put (i) gn+1 (x) − gn (x) :=

Z

X

φn+1 (y)µ(dy, U n |x) − φn (x),

where µ(A, U n |x) is the conditional measure of µ with respect to the σ-algebra U n . The existence fo such conditional measure is proved in §2.37. Evidently, gn+1 (x) − gn (x) is

24

Sergey V. Ludkovsky

the U n -measurable function. Multiplying this function on any bounded non-negative U n measurable function ψ(x) due to Equality 2.37(2) we deduce that Z

(ii) −

Z

X

X

[gn+1 (x) − gn (x)]ψ(x)µ(dx) = φn (x)ψ(x)µ(dx) =

Z

X

Z Z X

X

φn+1 (y)µ(dy, U n |x)ψ(x)µ(dx)

φn+1 (x)ψ(x)µ(dx) −

Z

X

φn (x)ψ(x)µ(dx).

Form this equality it follows, that (iii)

Z

X

[gn+1 (x) + φn+1 (x)]ψ(x)µ(dx) =

Z

X

[gn (x) + φn (x)]ψ(x)µ(dx),

since gn+1 (x) = ∑nk=1 [gk+1 (x) − gk (x)], with g1 (x) := 0. Let in Formula (ii) be ψ(x) = 1, when gn+1 (x) − gn (x) < 0 and ψ(x) = 0 in other cases. Then from the definition of the sub-martingale it follows that −

Z

X

ψ(x)µ(dx) =

Z

X

[gn+1 (x) − gn (x)]ψ(x)µ(dx) ≥ 0,

consequently, gn+1 (x) ≥ gn (x) µ-almost everywhere. 2.40. Remark. From the proof of the preceding lemma we get that Z

X

φn+1 (y)µ(dy, U n |x) ≥ φn (x)

(1)

for each n. In the martingale case we have here the equality instead of the inequality. If R |φ X n (x)|µ(dx) < ∞ for each n, then {φn : n} is the sub-martingale. If gn (x) are the same as in §2.39, then Z

X

gn (x)µ(dx) =

Z

X

[gn (x) − φn (x)]µ(dx) +

Z

X

φn (x)µ(dx).

Taking in 2.38(2) ψ(x) = 1 we get that X fnR(x)µ(dx) = X f1R(x)µ(dx) for the martingale { fn (x) = gnR(x) − φn (x) : n}.R Therefore, X gn (x)µ(dx) = X [g1 (x) − φ1 (x)]µ(dx) + R X φn (x)µ(dx) = X φn (x)µ(dx) − X φ1 (x)µ(dx). If for a sub-martingale the condition R

sup n

Z

is satisfied, then sup n

X

Z

X

R

|φn (x)|µ(dx) < ∞

(2)

gn (x)µ(dx) < ∞,

(3)

hence a non-decreasing sequence gn (x) has µ-almost everywhere a finite limit. Moreover, this linmit is µ-integrable. If {φn : n} is a super-martingale, then {−φn : n} is a submartingale and for it an integrable limit of the sequence −gn (x) exists. Thus for the proof of an existence of the limit lim φn (x) (4) n→∞

for martingales, sub-martingales and super-martingales the consideration of martingales only is sufficient.

Real-Valued Measures

25

2.41. Lemma. If {φn : n} is a martingale, then for all n and b > 0 the inequality µ({x : sup φk (x) ≥ b}) ≤ sup n

k

Z

φ+ n (x)µ(dx)/b

X

+ is satisfied, where φ+ n (x) = φn (x) for φn (x) ≥ 0, while φn (x) = 0 for φn (x) ≤ 0. R supn X φn (x)µ(dx) < ∞, then

µ({x : sup φk (x) ≥ b}) ≤ sup k

Z

X

φ+ n (x)µ(dx)/b.

(1) If

(2)

Proof. Let χk (x) = 1, when φ1 (x) < b,. . . ,φk−1 (x) < b, φk (x) ≥ 0, φk (x) = 0 in other cases, hence φk (x) is the A k -measurable function. Therefore, Z

X

φn (x)χk (x)µ(dx) =

= ··· = for each k ≤ n, consequently, Z

Z

X

X

φn−1 (x)χk (x)µ(dx)

φk (x)χk (x)µ(dx) ≥ b

n

X

Z

φn (x) ∑ χk (x)µ(dx) ≥ b k=1

Z

Z

X

χk (x)µ(dx)

n

∑ χk (x)µ(dx). X

(3)

k=1

On the other hand, the function ∑nk=1 χk (x) is characteristic of the set {x ∈ X : supk≤n φk (x) ≥ b} = Bn . From Equation 2.38(2) we infer that µ(Bn ) ≤

Z

Bn

φn (x)µ(dx)/b ≤

Z

X

φ+ n (x)µ(dx)/b.

This demonstrates Inequality (1). Taking the limit by n tending to the infinity we get (2) as well. 2.42. Theorem. If {φn (x) : n} is a martingale for which Condition 2.41(3) is satisfied, then µ-almost everywhere the limit 2.40(4) exists. Proof. We shall say that a sequence α1 , . . . , α2 , . . . crosses infinite times the segment [β1 , β2 ] with β1 < β2 , if it is possible to choose k1 < k2 < · · · so that αk1 ≥ β2 , αk2 ≤ β1 ,. . . , αk2n−1 ≥ β2 , αk2n ≤ β1 , . . . . Consider the set Bβ1 ,β2 consisting of all x ∈ X for which the sequence {φn (x) : n} crosses infinite number of times the segment [β1 , β2 ]. Denote by B− the set of all x ∈ X for which infn φn (x) = −∞, while B+ is the set of all x ∈ X with supn φn (x) = ∞. Therefore, the S set B+ ∪ B− ∪ [ β1 0. Then k2 (x) = j if k1 (x) < ∞ and φ p (x) > β1 and φ j (x) ≤ β1 for k1 (x) ≤ p < j. In others cases we put k2 (x) = ∞. If k2n (x) is defined, then k2n+1 (x) = j if k2n (x) < ∞ and φ p (x) < β2 while φ j (x) ≥ β2 for k2n (x) ≤ p < j. If either k2n (x) = ∞ or φ p (x) > β2 for p > k2n (x), then put

26

Sergey V. Ludkovsky

k2n+1 = ∞. If k2n+1 (x) is defined, then k2n+2 (x) = j. If k2n+1 (x) < ∞ and φ p (x) > β1 and φ j (x) ≤ β1 for k2n+1 (x) ≤ p < j, then k2n+2 (x) = j. If either k2n+1 (x) = ∞ or φ p (x) > β1 for p > k2n+1 (x), then k2n+2 (x) = ∞. We put further χn (x) = 1 when k2p (x) ≤ n < k2p+1 (x) for some p, while χn (x) = −1 when k2p−1 (x) ≤ n < k2p (x) for some p. Therefore, functions χn (x) are completely defined by φ1 (x), . . . , φn (x) so that χn is the Borel function of φ1 , . . . , φn . Thus χn is U n -measurable. Construct the functions n−1

gn (x) := φ1 (x) + ∑ [φk+1 (x) − φk (x)]χk (x). k=1

We shall show that {gn (x) : n} is the martingale. Denote by P n the σ-algebra generated by functions g1 (x), . . . , gn (x). Since gk (x) is U k -measurable, then P n ⊂ U n for each n. If a function ψ(x) is bounded and P n -measurable ,then it is U n -measurable. This implies that Z

=

Z

X

X

gn+1 (x)ψ(x)µ(dx)

[gn (x) + (φn+1 (x) − φn (x))χn (x)]ψ(x)µ(dx) =

Z

X

gn (x)ψ(x)µ(dx),

since X [φn+1 (x) − φn (x)]χn (x)ψ(x)µ(dx) = 0 due to U n -measurability and boundedness of χn (x)ψ(x), consequently, gn (x) is the martingale. From g1 (x) = φ1 (x) it follows that R

Z

X

gn (x)µ(dx) =

Z

X

g1 (x)µ(dx) =

Z

X

φ1 (x)µ(dx).

For each n ≤ k1 (x) we have gn (x) = φn (x), while for each k1 (x) < n ≤ k2 (x) there is the equality gn (x) − gk1 (x) (x) = φk1 (x) (x) − φn (x) or gn (x) = 2φk1 (x) (x) − φn (x) ≥ 2β2 − φn (x). This means that gk2 (x) (x) ≥ 2β2 − β1 > β1 ≥ φk2 (x) (x) and gn (x) ≥ φn (x) for each k2 (x) < n ≤ k3 (x). When the inequalities k3 (x) < n ≤ k4 (x) are satisfied, then gn (x) − gk3 (x) (x) = φk3 (x) (x) − φn (x) also gn (x) ≥ 2φk3 (x) (x) − φn (x) ≥ 2β2 − φn (x). So we deduce that gn (x) ≥ min[φn (x), 2β2 − φn (x)].

(1)

n

− + − + − Consider now the functions φn = φ+ n − φn , gn = gn − gn , where gn (x)gn (x) = 0, + − + − = 0, φn (x) ≥ 0, φn (x) ≥ 0, gn (x) ≥ 0, gn (x) ≥ 0. From Inequality (1) it follows that − + g− n (x) = max[gn (x), gn (x) − 2β2 ] ≤ |φn (x)| + 2|β2 |.

− φ+ n (x)φn (x)

Therefore,

≤

Z

Z X

X

g+ n (x)µ(dx) =

Z

X

gn (x)µ(dx) +

[φn (x) + |φn (x)| + 2|β2 |]µ(dx) ≤ 2

Z

Z X

X

g− n (x)µ(dx)

g+ n (x)µ(dx) + 2|β2 |,

consequently, µ({x ∈ X : sup gn (x) = ∞}) = lim µ({x ∈ X : sup gn (x) ≥ b}) n

b→∞

n

Real-Valued Measures ≤ lim sup b→∞ n

Z

X

27

g+ n (x)µ(dx)/b = 0.

We have that kn (x) < ∞ for each n when x ∈ Bβ1 ,β2 , hence Bβ1 ,β2 ⊂ {x ∈ X : supn gn (x) = ∞}. If kn (x) < ∞, then gkn (x) (x) − gkn−1 (x) (x) ≥ β2 − β1 and gk1 (x) (x) ≥ β2 , consequently, gkn (x) (x) ≥ β2 + (n − 1)[β2 − β1 ] and inevitably supn gn (x) = ∞ for x ∈ Bβ1 ,β2 . Thus µ(Bβ1 ,β2 ) = 0. 2.43. Corollary. Each non-negative martingale {φn (x) : n} has a limit almost everywhere by the measure µ. If φ(x) = limn→∞ φn (x) (mod µ), then Z

X

Z

φ(x)µ(dx) ≤

φ1 (x)µ(dx).

X

(1)

Proof. For a non-negative martingale X |φn (x)|µ(dx) = X φn (x)µ(dx) = X φ1 (x)µ(dx), consequently, Condition 2.40(3) is satisfied. Then Inequality (1) follows from the Fatou Theorem II.6.2 [Shir89]. 2.44. Corollary. If {φn : n} is a non-negative sub-martingale and R

R

sup n

Z

X

R

φn (x)µ(dx) < ∞,

then the limit φ(x) = lim φn (x) (mod µ) n→∞

exists and

Z

X

φ(x)µ(dx) ≤ sup n

Z

X

(1)

φn (x)ψ(x)µ(dx).

(2)

φn (x)ψ(x)µ(dx)

(3)

If in (2) the equality is, then Z

X

φ(x)ψ(x)µ(dx) ≤

Z

X

for each U n -measurable bounded non-negative function ψ. Proof. In view of the Fubini theorem (see Theorem 8 in §II.6 [Shir89]) we have Z

X

φ(x)ψ(x)µ(dx) ≤ lim

Z

φm (x)ψ(x)µ(dx).

(4)

Z

φm (x)[b − ψ(x)]µ(dx)

(5)

m→∞ X

For each b > ψ(x) the inequality Z

X

φ(x)[b − ψ(x)]µ(dx) ≤ lim

m→∞ X

is satisfied. If in eitherR (4) or (5) would be the strict inequality, then the inequality X φ(x)µ(dx) < limm→∞ X φm (x)µ(dx) would be satisfied. This would contradict that in (2) is the equality. For martingales under the same conditions in (3) is the equality. 2.45. Note. If {φn : n} is a martingale and g is a non-negative convex (downward) function defined on (−∞, ∞) so that g(φn (x))µ(dx) < ∞ for each n ∈ N, then {g(φn (x)) : n} is the sub-martingale. Indeed, in view of Remark 2.40 and the Jensen’s inequality (see §II.6 [Shir89])  Z Z Z g(φn+1 (y))µ(dy, U n |x) ψ(x)µ(dx) g(φn+1 (x))ψ(x)µ(dx) = R

X

X

X

28 ≥

Z

g X

Z

Sergey V. Ludkovsky  φn+1 (y)µ(dy, U n |x) ψ(x)µ(dx)

X

=

Z

X

g(φn (x))ψ(x)µ(dx)

(1)

for each non-negative U n -measurable bounded function ψ. If g(t) is non-decreasing and {φn (x) : n} is a sub-martingale, then the relations above with the inequality ≥ instead of the latter equality = imply that {g(φn (x)) : n} is the sub-martingale. In the process of the proof of Theorem 2.42 we have demonstrated that µ({x ∈ X : kn (x) ≤ m}) = µ({x ∈ X : sup gn (x) > β2 + (n − 1)(β2 − β1 )}) n≤m

 Z + φm (x)µ(dx) + 2|β2 |]/[β2 + (n − 1)(β2 − β1 ) . ≤2

(2)

X

The set {x ∈ X : kn (x) ≤ m} coincides with the set of those x for which the sequence φ1 (x), . . . , φm (x) not less than n times intersects the segment [β1 , β2 ]. That is for each n there exist a sequence k1 < k2 < · · · < kn ≤ m with φk1 (x) ≥ β2 , φk2 (x) ≤ β1 , φk3 (x) ≥ β2 and so on.

1.3.

Quasi-invariant Measures

In this section after few preliminary statements there are given the definition of a quasiinvariant measure and the theorems about quasi-invariance of measures relative to transformations of a Banach space X over K. S 3.1. Let X be a Banach space over K, (Ln : n) be a sequence of subspaces, cl( n Ln ) = j X, Ln ⊂ Ln+1 for each n, µ j be probability measures, µ2 ≪ µ1 , (µLn ) be sequences of weak distributions, also let there exist derivatives ρn (x) = µ2Ln (dx)/µ1Ln (dx) and the following limit ρ(x) := lim ρn (x) n→∞

(1)

exists. Theorem. If µ j are real-valued probability measures and in addition Z

(i)

X

ρ(x)µ1 (dx) = 1

with ρ ∈ L1 (µ1 ), then this is equivalent to the following: there exists (ii) ρ(x) = µ2 (dx)/µ1 (dx) (mod µ1 ). Proof. Let µ2 ≪ µ1 and π(x) = dµ2 (x)/dµ1 (x). Denote by µkn (∗|x) the conditional measure of µk relative to the σ-algebra BLn . Therefore, for each BLn -measurable non-negative bounded function ψ(x) there are satisfied the equalities: Z

ψ(x)µ (dx) = 2

X

Z

X

ψ(x)π(x)µ1 (dx)

Real-Valued Measures  Z Z 1 = ψ(x) π(y)µn (dy|x) µ1 (dx).

29

X

X

On the other hand, Z

ψ(x)µ (dx) = 2

X

Z

X

ρn (x) =

ρn (PLn x)ψ(x)µ1 (dx), hence Z

(2)

π(y)µ1n (dy|x).

X

We shall demonstrate that ρn (x) is uniformly relative to n integrable by the measure µ1 . For this we have to show that for each b > 0 there exists β so that Z

{x∈X: ρn (x)>β}

ρn (x)µ1 (dx) < b.

(3)

Define the function gβ (t) = 0 for β > t and gβ (t) = t − β for β ≤ t. Then Relation (3) is equivalent with Z

X

gβ (ρn (x))µ1 (dx) + βµ1 ({x ∈ X : ρn (x) > β}) < b.

From gβ (ρn (x)) = gβ we infer that

Z

Z

X



π(y)µ1n (dy|x)

1

X

gβ (ρn (x))µ (dx) ≤

Z

X

βµ ({x ∈ X : ρn (x) > β}) ≤ 2 Z

X

X

gβ (π(y))µ1n (dy|x)

gβ (π(y))µ1 (dy). Moreover,

1

≤2

≤

Z

(4)

Z

X

gβ/2 (ρn (x))µ1 (dx)

gβ/2 (π(x))µ1 (dx).

Therefore, Conditions (3, 4) will be satisfied as soon as the parameter β is chosen such that 1 X gβ/2 (π(x))µ (dx) < b/3. From theRuniform relative to n integrability of ρn (x) by the measure µ it follows that in the equality X ρn (x)µ1 (dx) = 1 we can take the limit by n → ∞ under the integral. This demonstrates (i). For each A ∈ BLn in the relations

R

µ2 (A) = lim

Z

n→∞ X

χA (x)ρn (x)µ1 (dx) =

Z

X

χA (x)ρ(x)µ1 (dx)

we can take the limit with n tending to the infinity under the integral. This proves (ii). Suppose now that (i) is satisfied. Then from Corollary 2.44 we deduce that Z

ρ(x)ψ(x)µ (dx) = 1

X

Z

ρn (x)ψ(x)µ (dx) = 1

X

Z

X

ψ(x)µ2 (dx)

for every BLn -measurable bounded non-negative function ψ(x). Thus Z

A

ρ(x)µ1 (dx) =

Z

µ2 (dx) = µ2 (A) A

(5)

30

Sergey V. Ludkovsky

is satisfied for each A ∈ BLn . The class of all functions for which (5) is accomplished contains BLn and is monotone, consequently, it contains also the minimal σ-algebra containing each BLn , hence it contains the entire σ-algebra B f (X). From (5) for each A ∈ B f (X) it follows that µ2 is absolutely continuous relative to µ1 , µ2 ≪ µ1 , and Formula (ii) is satisfied. 3.2. Theorem. Probability measures µ j : B f (X) → R, j = 1, 2, for a Banach space X over K are orthogonal µ1 ⊥ µ2 if and only if ρ(x) = 0 (mod µR1 ). Proof. If A ∈ BLn , then for each n < m we have µ2 (A) = A ρm (x)µ1 (dx). In view of the Fatou theorem taking the limit with m tending to the infinity gives µ2 (A) ≥

Z

A

ρ(x)µ1 (dx).

(1)

This spreads on all A ∈ B f (X). Suppose now that µ2R⊥ µ1 , hence there exists a set A so that µ2 (A) = 0 and µ1 (X \A) = 0. From (1) we infer that A ρ(x)µ1 (dx) = 0. Since µ1 (X \ A) = 0, then ρ(x) = 0 (mod µ1 ). This demonstrates the necessity. Let now ρ(x) = 0 (mod µ1 ). We shall show that µ1 ⊥ µ2 . Suppose the contrary. So we can present µ2 in the form µ2 = βν1 + (1 − β)ν2 , where ν1 ≪ µ1 and ν2 ⊥ µ1 , 0 < β ≤ 1. Put ρ1n (x) := dν1Ln (PLn x)/dµ1Ln (PLn x). In accordance with Theorem 3.1 there exists the limit lim ρ1 (x) n→∞ n

= dν1 (x)/dµ1 (x) (mod µ1 ).

But βρ1n (x) ≤ ρn (x), consequently, limn→∞ ρ1n (x) ≤ limn→∞ ρn (x)/b = 0, that is, dν1 (x)/dµ1 (x) = 0 (mod µ1 ) contradicting the supposition about the absolute continuity of ν1 relative to µ1 , hence µ1 ⊥ µ2 . 3.2.1. Theorem. The function ρ(x) defined by Relation 3.1(2) is the density of an absolute continuous part of the measure µ2 relative to the measure µ1 , so that Formula 3.1 (ii) is accomplished in all cases. Proof. Let µ2 = βν1 + (1 − β)ν2 with ν1 ≪ µ1 and ν2 ⊥ µ1 and 0 ≤ β < 1. Denote by 1 νLn and ν2Ln projections of measures ν1 and ν2 on Ln , also ρ1n (x) = dν1Ln (PLn x)/dµ1Ln (PLn x), ρ2n (x) = dν2 (PLn x)/dµ1Ln (PLn x). Then ρn (x) = βρ1n (x) + (1 − β)ρ2n (x). In accordance with Theorem 3.1 the limit limn→∞ ρ1n (x) = dν1 (x)/dµ1 (x) (mod µ1 ) exists. By Theorem 3.2 limn→∞ ρ2n (x) = 0 (mod µ1 ), hence limn→∞ ρn (x) = βdν1 (x)/dµ1 (x) = dµ2 (x)/dµ1 (x) (mod µ1 ). 3.2.2. Theorem. Let µ1 and µ2 be arbitrary Borel probability measures on a Banach space X over K, let also Ln be an increasing sequence of K-vector subspaces in X so that S n Ln is dense in X. Then dµ2 (x)/dµ1 (x) = lim dµ2Ln (PLn x)/dµ1Ln (PLn x) n→∞

(mod µ1 ),

(1)

while µ2 ≪ µ1 if and only if X [dµ2 (x)/dµ1 (x)]µ1 (dx) = 1. Proof. Construct the decomposition µ2 = ν1 + ν2 such that ν2Ln ≪ µ1Ln for each n ∈ N and Z (2) lim [dν1Ln (x)/dµ1Ln (x)]µ1Ln (dx) = 0. R

n→∞ X

Consider a set An ∈ B f (Ln ) such that µ1Ln (An ) = 0 and the measure µ2 (B \ An ) is absolutely continuous relative to µ1Ln (B) on B f (Ln ). Denote by En the cylinder set in X with the base

Real-Valued Measures

31

2 2 1 2 2 2 An . Put E = ∞ n=1 En , ν (B) = µ (B \ E), ν (B) = µ (B ∩ E). Then νLn (A) ≤ µLn (A \ An ), consequently, ν2Ln ≪ µ1Ln by the construction of the set An . From µ1Ln (PL−1 [Ak ∩ Ln ]) = 0 for n all k ≤ n and from the fact that dν1Ln /dµ1Ln is different from zero only on the set PLn E we get

S

Z

X

n i  h [ Ek , [dν1Ln (x)/dµ1Ln (x)]µ1Ln (dx) = ν1Ln PLn E \ k=1

since A [dν(x)/dµ(x)]µ(dx) ≤ ν(A) for any non-negative bounded measures. S Consider the cylinder set Cn from BLn with the base PLn [E \ nk=1 Ak ]. Evidently, Cn ⊃ T T / then ∞ Cn+1 and limn→∞ ν1 (Cn ) = limm→∞ ν1 ( m Since En ∩ Cn = 0, n=1 Cn ). n=1 Cn has the T S 1 1( E = E also. Thus ν void intersection with each Ek and hence with n Cn ) = ν (E ∩ k k R T [ n Cn ]) = 0. From the inequality X [dν1 (x)/dµ1 (x)]µ1Ln (dx) ≤ ν1 (Cn ) Equality (2) follows. For the measure ν2 the statements of Theorems 3.1-3.2.1 are accomplished. From the T construction of ν1 for each x in X outside the set [ n Cn ] ∪ E we have R

lim dν1Ln (PLn x)/dµ1Ln (PLn x) = 0.

(3)

n→∞

The latter equality is satisfied µ1 -almost everywhere, since ν1 ( n Cn ) = 0 and µ1 (E) = 0. Therefore, lim dµ2Ln (PLn x)/dµ1Ln (PLn x) = lim dν2Ln (PLn x)/dµ1Ln (PLn x) T

n→∞

n→∞

= dν2 (x)/dµ1 (x) = dµ2 (x)/dµ1 (x) (mod µ1 ). 3.2.3. Theorem. Let the measurable Banach space (X, B (X)), where X is over K and (X, B (X)) = (X1 , B (X1 )) × (X2 , B (X2 )). Suppose that two measures ν j and µ j are given on (X j , B (X j )) for j = 1, 2. For measures µ = µ1 × µ2 and ν = ν1 × ν2 on X, B (X)) the relation ν ≪ µ is satisfied if and only if ν1 ≪ µ1 and ν2 ≪ µ2 . If this is the case, then dν(x)/dµ(x) = [dν1 (P1 x)/dµ1 (P1 x)][dν2 (P2 x)/dµ2 (P2 x)],

(1)

where P1 x = x1 ∈ X1 and P2 x = x2 ∈ X2 , x = (x1 , x2 ). Proof. At first prove the necessity. If ν ≪ µ and µ(A1 × X2 ) = µ1 (A1 ) = 0, then ν(A1 × X2 ) = ν1 (A1 ) = 0 for each A1 ∈ B (X1 ). Therefore, ν1 ≪ µ1 . Analogously ν2 ≪ µ2 . Let now ν j ≪ µ j for j = 1, 2. Denote by F 0 the algebra of sets from Sn B (X) having the form k=1 (A1k × A2k ), where Akj ∈ B (X j ), k, n ∈ N. Let ρ(x) = [dν1 (P1 x)/dµ1 (P1 x)][dν2 (P2 x)/dµ2 (P2 x)]. From Z

A1 ×A2

ρ(x)µ(dx) =

Z

A1

[dν1 (x1 )/dµ1 (x1 )]µ1 (dx1 )

Z

A2

[dν2 (x2 )/dµ2 (x2 )]µ2 (dx2 )

= ν1 (A1 )ν2 (A2 ) = ν(A1 × A2 ) for each A ∈ F 0 the equality Z

A

ρ(x)µ(dx) = ν(A)

(2)

follows. The latter relation is satisfied on a monotone class containing F 0 , hence it is satisfied for all A ∈ B (X). This implies ν ≪ µ and Formula (1).

32

Sergey V. Ludkovsky

3.2.4. Theorem. If probability measures µ1 and µ2 are defined on a measurable space (X, B (X)), where X is a Banach space over K, µ2 ≪ µ1 , ν j (C) := µ j ( f −1 (C)), ∀C ∈ B (Y )),

(1)

f : (X, B (X)) → (Y, B (Y )) is a measurable mapping, then Z

dν2 (y)/dν1 (y) =

X

[dµ2 (x)/dµ1 (x)]µ1 (dx, B 1 | f −1 (y)),

(2)

where µ1 (∗, B 1 |z) is the conditional measure of µ1 relative to the σ-algebra B 1 generated by sets of the form f −1 (C), C ∈ B (Y ). Proof. Show at first that µ1 (A, B 1 |z) for A ∈ B (X) is constant on the inverse image −1 f (y) for any f . Consider Ay := {z : f (z) = y}, Ay ∈ B 1 . For each B 1 -measurable set A1 either Ay ⊂ A1 or Ay = X \ A1 . Therefore, each B 1 -measurable function is constant on Ay . The function µ1 (A, B 1 |z) of z is B 1 -measurable, hence µ1 (A, B 1 | f −1 (y)) is independent from a choice of a point in the inverse image f −1 (y) of a point y. Let ψ(y) be a bounded B (Y )-measurable function on (Y, B (Y )). Then the function ψ( f (x)) = φ(x) is B 1 -measurable and bounded on (X, B (X)). Thus Z

ψ(y)ν (dy) = 2

X

Z

ψ( f (x))µ (dx) = 2

X

Z

X

ψ( f (x))[dµ2 (x)/dµ1 (x)]µ1 (dx).

Expression 2.37(2) implies that Z

X

Z

ψ( f (x))[dµ (x)/dµ (x)]µ (dx)= 2

Put

1

Z

X

1

X

ψ( f (z))

Z

X

[dµ2 (x)/dµ1 (x)]µ1 (dx, B 1 |z)µ1 (dz).

[dµ2 (x)/dµ1 (x)]µ1 (dx, B 1 | f −1 (y)) = ρ1 (y),

(3)

2 1 1 1 1 then X ψ(y)ν (dy) X ψ( f (x))ρ ( f (z))µ (dz) = X ψ(y)ρ (y)ν (dy), consequently, R 1 1 = X ψ(y)ρ (y)ν (dy), that demonstrates this theorem. 3.2.5. Corollary. Let conditions of Theorem 3.2.4 be satisfied and in addition on (X, B (X)) be two other measures ν1 and ν2 be given such that ν j ≪ µ j , j = 1, 2. Then ν1 × ν2 ≪ µ1 × µ2 and [d(ν1 × ν2 )/d(µ1 × µ2 )](x1 , x2 ) = [dν1 /dµ1 ](x1 )[dν2 /dµ2 ](x2 ). Moreover, ν1 ∗ ν2 ≪ µ1 ∗ µ2 and

R

R

R

[d(ν1 ∗ ν2 )/d(µ1 ∗ µ2 )](x) =

Z

X

(1)

[dν1 /dµ1 ](x1 )[dν2 /dµ2 ](x2 )(µ1 ∗ µ2 )(dx1 × dx2 , B ∗ |η−1 (x)),

where η(x1 , x2 ) = x1 + x2 , B ∗ denotes the σ-algebra generated by the sets η−1 (A), A ∈ B (X). R Proof. As above the convolution of measures is µ1 ∗ µ2 (A) = X µ2 (A − x)µ1 (dx), where A ∈ B (X), A − x := {y ∈ X : y + x ∈ A}. The set SA = {(x1 , x2 ) : x1 + x2 ∈ A} is B (X) × B (X)-measurable for B -measurable A. Put SxA1 := {x2 ∈ X : (x1 , x2 ) ∈ SA } is the section of SA by the first coordinate. On the other hand, [µ1 × µ2 ](SA ) =

Z

X

µ2 (SxA1 )µ1 (dx1 ) =

Z

X

µ2 (A − x1 )µ1 (dx1 )

(2)

Real-Valued Measures =

Z

µ1 (SxA2 )µ2 (dx2 ) =

X

Z

X

33

µ1 (A − x2 )µ2 (dx2 )

. This implies that the convolution is commutative. The convolution is obtained from the product of measures with the help of the mapping η(x1 , x2 ) = x1 + x2 . So this corollary follows from Theorems 3.2.3 and 3.2.4. 3.3. Note. For real-valued measures µ j on B f (X) for a Banach space X over the infinite locally compact field K with the non-trivial multiplicative non-Archimedean norm there is the important Kakutani’s theorem formulated in §3.3.1 below. Its proof is given in Theorem 4.1 §II.4.6[DF91] and in [Kak48] for abstract measurable spaces. Let µ and ν be two probability measures in a measurable space (X, B ) and let λ be a probability measure in (X,RB ) such that µ and ν are absolutely continuous relative to λ. Then the integral H(µ, ν) := X (dµ/dλ)1/2 (dν/dλ)1/2 dλ is called the Hellinger integral of µ and ν. It has the properties: (i) 0 ≤ H(µ, ν) ≤ 1; (ii) H(µ, ν) = 1 is equivalent to µ = ν; (iii) H(µ, ν) = 0 if and only if µ is orthogonal to ν; (iv) if µ is equivalent to ν, then H(µ, ν) > 0. 3.3.1. Theorem. Let µn and νn be two sequences of probability measures in (Xn , B n ) N∞ ∞ ˜ ˜ (see above), define µ = ∏∞ n=1 B n . If µn is n=1 µn , ν = ∏n=1 νn in (X, B ), where B := equivalent to νn for each n ∈ N, then µ and ν are either equivalent or orthogonal depending on whether ∏∞ n=1 H(µn , νn ) converges to a finite positive number or diverges to zero, where ∞ dµ/dν = ∏n=1 dµn /dνn when µ and ν are equivalent. 3.4. Theorem. Let ν, µ, ν j , µ j be probability measures with values in R, X = ∏∞j=1 X j be a product of completely regular spaces X j with the small inductive dimension ind(X j ) = 0. Then ν ≪ µ if and only if two conditions are satisfied: (a) ν j ≪ µ j for each j and (b) ∏∞j=1 β j converges to β, ∞ > β > 0, where β j := k(ρ j )1/2 kL1 (X j ,µ j ) , ρ j (x) = ν j (dx)/µ j (dx). Proof. The necessity of (a) is evident. If ν ≪ µ, then ∞

dν(x)/dµ(x) = ∏ dνk (Pk x)/dµk (Pk x). k=1

Therefore,

n

dν(x)/dµ(x) = lim ∏ dνk (Pk x)/dµk (Pk x). n→∞

(1)

k=1

The functions gn (x)R := ∏nk=1 [dνk (Pk x)/dµk (Pk x)]1/2 are uniformly relative to n integrable, since the integrals X [gn (x)]2 µ(dx) = 1 are uniformly bounded. Combining this with (1) implies Z

1/2

[dν(x)/dµ(x)]

µ(dx) = lim

n→∞

X n

= lim ∏ n→∞

Z

k=1 X

Z

n

[dνk (Pk x)/dµk (Pk x)]1/2 µ(dx) ∏ X k=1

[dνk (Pk x)/dµk (Pk x)]1/2 µ(dx),

34

Sergey V. Ludkovsky

hence (b) is proved. Prove now the sufficiency of conditions of this theorem. In view of Theorem 3.3.1 under the condition (a) there is the alternative either ν ≪ µ or ν ⊥ µ. The latter will be, if ∞ ∑∞ k=1 µk (Bk ) = ∞ or ∏k=1 dνk (Pk x)/dµk (Pk x) = 0 (mod µ). This means that it is sufficient to show that if ν ⊥ µ and (a) is satisfied, then the infinite product in (b) diverges to zero. In this case limn→∞ ∏∞ k=1 dνk (Pk x)/dµk (Pk x) = 0 (mod µ), consequently, n

lim ∏ [dνk (Pk x)/dµk (Pk x)]1/2 = 0.

n→∞

(2)

k=1

Using the uniform integrability we infer that Equality (2) can be integrated interchanging the operations of integration and taking of the limit. Thus lim

Z

n

∏[dνk (Pk x)/dµk (Pk x)]

1/2

n

µ(dx) = lim ∏ n→∞

n→∞ X k=1

Z

k=1 X

[dνk (Pk x)/dµk (Pk x)]1/2 µ(dx).

3.5. Definition. Let X be a Banach space over K, Y be a completely regular space with the small inductive dimension ind(X) = 0, ν : E → R, µy : B → R for each y ∈ Y , suppose µy (A) ∈ L1 (Y, ν, R) for each A ∈ B , where B is a σ-algebra on X and E is a σ-algebra on Y . Then we define: Z (i) µ(A) = µy (A)ν(dy). Y

A measure µ is called mixed. We define measures π j by the formula: (ii) π (A ×C) = j

Z

µ j,y (A)ν j (dy),

C

where j = 1, 2 and µy, j together with ν j are defined as above µy and ν. 3.5.1. Theorem. If π j , µ j,y and ν j are defined as in §3.5, j = 1, 2, X is of separable type over the locally compact field K, where µ j,y and ν j are bounded non-negative measures with 0 < ν j (Y ) < ∞ and 0 < µ j,y (X) < ∞ for each j and y, then (i) if π2 ≪ π1 , then ν2 ≪ ν1 and µ2,y ≪ µ1,y for ν2 -almost all y; (ii) if ν2 ≪ ν1 and µ2,y ≪ µ1,y for ν2 -almost all y, then π2 ≪ π1 . Moreover, there exists a B × E -measurable function η(y, x) = dµ2,y (x)/dµ1,y (x) so that dπ2 (x, y)/dπ1 (x, y) = [dν2 (y)/dν1 (y)]η(y, x).

(1)

Proof. Let π2 ≪ π1 . We put ρ(y, x) = dπR2 (x, y)/dπ1 (x, y). For each B timesE measurable bounded function φ(x, y) the function X φ(x, y)µk,y (dx) is E -measurable and  Z Z Z φ(x, y)µ j,y (dx) ν j (dy). (2) φ(x, y)π j (dx, dy) = X×Y

X

Y

Therefore, for each B ∈ B and C ∈ E this implies that π2 (B ×C) = =

Z Z  C B

ρ(y, x)µ (dx)/ 1,y

Z

Z Z

C B

ρ(y, x)µ1,y (dx)ν1 (dy)

  Z 1,y ρ(y, z)µ (dz) ν1 (dy). ρ(y, z)µ (dz) 1,y

X

X

Real-Valued Measures

35

Taking B = X we get π2 (X ×C) = ν2 (C) =

Z Z C

hence ν2 ≪ ν1 and dν2 (y)/dν1 (y) =

Z

X

X

 ρ(y, x)µ1,y (dx) ν1 (dy, ρ(y, x)µ1,y (dx)

and inevitably π (B ×C) = 2

where

Z

2,y

2

µ (B)ν (dy) =

C

Z Z C

 η(y, x)µ (dx) ν2 (dy), 1,y

B

−1 Z 1,y . ρ(y, x)µ (dx) η(y, x) = ρ(y, x)

(3)

X

Thus 2,y

µ (B) =

Z

B

η(y, x)µ1,y (dx)

(4)

for ν2 -almost all y and each B ∈ B . Choose a sequence of sets Bk ∈ B such that it will form an algebra generating B . This is possible, since the field K is separable and X is of separable type over X with a countable base of neighborhoods of zero. Therefore, there exists a set F ∈ E so that ν2 (F) = ν2 (Y ) and µ2,y (Bk ) =

Z

Bk

η(y, x)ν1,y (dx)

for each y ∈ F and all k, consequently, (4) is satisfied for all y ∈ F and B ∈ B . For proving (ii) we first show that Rπ2 ≪ π1 . Consider A ∈ B × E , Ay = {x : (x, y) ∈ A}, so from (2) it follows that π j (A) = Y µ j,y (Ay )ν j (dy). If π1 (A) = 0, then µ1,y (Ay ) = 0 (mod ν1 ), hence µ2,y (Ay ) = 0 (mod ν2 ), since ν2 ≪ ν1 and µ2,y ≪ µP 1, y for ν2 -almost all y. Thus π2 (A) = 0, that is π2 ≪ π1 . For proving the existence of η and demonstrating (1) use the proof of (i) defining η by Formula (3). 3.5.2 Corollary. If conditions of the Theorem 3.5.1 are satisfied and ν2 ≪ ν1 and 2,y µ ≪ µ1,y for ν2 -almost all y, then µ2 ≪ µ1 and dµ2 (x)/dµ1 (x) =

Z

[dν2 (y)/dν1 (y)][dµ2,y (x)/dµ1,y (x)]π1 (dy|x),

(1)

X

where π1 (C|x) is the conditional measure defined by the equation: Z

B

π1 (C|x)µ1 (dx) = π1 (B ×C).

(2)

Proof. This follows from Theorems 3.2.4 and 3.5.1. Formula (1) is the consequence of 3.2.4(2), since f ((x; y)) = x in the considered situation. 3.5.3 Lemma. Let conditions of Theorem 3.5.1 be satisfied. If in addition

36

Sergey V. Ludkovsky (i) there exists a measure λ such that µ1,y ≪ λ for all y, then dµ2 (y)/dµ1 (y) =

Z

Y

η(y, x)[dµ1,y (x)/dλ(x)]ν2 (dy)/

Z

[dµ1,y (x)/dλ(x)]ν1 (dy);

(1)

Y

(ii) all measures µ1,y for different y are orthogonal to each other for different values of y, moreover, there exist such pairwise non-intersecting By ∈ B so that µ1,y (By ) = 1 and µ1,y (Bz ) = 0 for z 6= y and the function ρ(x) = dµ2,y (x)/dµ1,y (x) with x ∈ By is B measurable, also ν1 = ν2 , then dµ2 (x)/dµ1 (x) = ρ(x). Proof. (i). Let ζ(x, y) = dµ1,y (x)/dλ(x). We can choose ζ(x, y) to be B × E -measurable using Theorem 3.5.1 and considering besides the measure π1 also the measure u = λ × ν1 . Then Z Z π1 (B ×C) = ζ(x, y)λ(dx)ν1 (dy), B C

Z Z

µ (B) = π (B ×Y ) = 1

1

B Y

consequently, µ1 ≪ λ and

dµ1 (x)/dλ(x) =

Z

Y

ζ(x, y)ν1 (dy)λ(dx),

ζ(x, y)ν1 (dy).

(2)

Evidently, µ2,y ≪ λ and dµ2,y (x)/dµ1,y (x) = η(y, x)ζ(x, y). We find also 2

dµ (x)/dλ(x) =

Z

Y

From µ2 ≪ µ1 ≪ λ we deduce that

η(y, x)ζ(x, y)ν2 (dy).

(3)

dµ2 /dµ1 = [dµ2 /dλ][dµ1 /dλ]−1 , hence from (2, 3) Equation (1) follows. To demonstrate (ii) mention that Z

φ(x)ρ(x)µ (dx) = 1

X

=

Z Z Y

=

Z Z Y

Y

 φ(x)ρ(x)µ (dx) ν1 (dy) 1,y

X

 φ(x)[dµ (x)/dµ 1, y(x)]µ (dx) ν1 (dy) 2,y

X

Z Z

1

1,y

 Z φ(x)µ (dx) ν1 (dy) = φ(x)µ2 (dx) 2,y

X

X

for each bounded B -measurable function φ(x). 3.5.4. Theorem. Let the families of measures µ j,y , j = 1, 2, on (X, B ) satisfy conditions: (i) µ j,y (B) is E -measurable for each B ∈ E ; (ii) there exist such B -measurable sets By so that µ j,y (Bz ) = 1 for y = z and µ j,y (Bz ) = 0 S for z 6= y and y∈C By ∈ B for each C ∈ E , By ∩ Bz = 0/ for each y 6= z, while ν j is a measure

Real-Valued Measures

37

on (Y, E ) for j = 1, 2. Suppose that measures µ j are defined by Equation 3.5(i) with µ j,y and ν j instead of µy and ν respectively, where µ j,y and ν j are bounded non-negative measures with 0 < ν j (Y ) < ∞ and 0 < µ j,y (X) < ∞ for each j and y. Then µ2 ≪ µ1 if and only if: (iii) ν2 ≪ ν1 and (iv) µ2,y ≪ µ1,y for ν2 -almost all y, if so, then dµ2 (x)/dµ1 (x) = [dµ2,y (x)/dµ1,y (x)][dν2 (x)/dν1 (x)]

(1)

for each x ∈ By . Proof. The mapping f ((x; y)) = x admits almost everywhere by the measure inversion S (x; y) = g(x), where g(x) = (x; y) for x ∈ By . Thus g(x) is defined on G := G(C) := y∈C By , T where C ∈ E . Consider the algebra ϒ which is the intersection y B µ1,y of completions B µ1,y of the algebra B by the measures µ1,y . Then G ∈ ϒ. If λ1 is the completion of the measure µ1 , then G is λ1 -measurable and λ1 (G) = 1. Choose By so that G(C) ∈ B for each C ∈ E . Then g(x) is B -measurable and {x : g(x) ∈ B ×C} = (G(C) ∩ B ∈ B . Consider as well the measure v on B × E which is the image of the measure µ1 under the mapping g such that 1

v(B ×C) = µ (G(C) ∩ B) =

Z

µ1,z (G(C) ∩ B)ν1 (dz)

Y

=

Z

Y

µ1,z (Bz ∩ B)χC (z)ν1 (dz) =

Z

Y

µ1,z (B)χC (z)ν1 (dz) = π1 (B ×C),

since µ1,z (G(C) ∩ B) = µ1,z (Bz ∩ [G(C) ∩ B]) = χC (y)µ1,z (B), where χC (y) is the characteristic function of the set C. R Prove now Formula (1). For this introduce the measure w1 (B):= Yµ1,y (B)ν2 (dy). From Statement (ii) of Lemma 3.5.3 we know that dµ2 (x)/dµ1 (x) = dµ2,y (x)/dµ1,y (x) for each x ∈ By . For each bounded B -measurable function φ with ρ(x)=dν2 (x)/dν1 (x) for each x ∈ By there are the equalities   Z

=

Z Z Y

φ(x)µ (dx) = 1

X

Z

Y

Z

X

φ(x)µ1,y (dx) ν2 (dy)

  Z Z 1,y 2 1 1 φ(x)ρ(x)µ (dx) ν1 (dy) φ(x)µ (dx) [dν (y)/dν (y)]ν (dy) = 1,y

X

Y

=

Z

X

X

φ(x)ρ(x)µ1 (dx),

hence dw1 (x)/dµ1 (x) = ρ(x). Finally applying dµ2 /dµ1 = [dµ2 /dw1 ][dw1 /dµ1 ] implies (1). 3.6. Definition. For a Banach space X over a locally compact infinite field K with a non-trivial non-Archimedean multiplicative norm an element a ∈ X is called an admissible shift of a measure µ, if µa ≪ µ, where µa (A) = µ(S−a A) for each A in B f (X), Sa A := a + A, ρ(a, x) := ρµ (a, x) := µa (dx)/µ(dx), Mµ := [a ∈ X : µa ≪ µ] (see §§2.1 and 2.36). A vector a ∈ X is called a partially admissible shift for the measure µ, if dµa /dµ is not identical to zero relative to the measure µ, that is µa contains a component absolutely continuous relative to µ. In such situation denote ρ˜ µ (a, x) = dµa (x)/dµ(x) the density of the

38

Sergey V. Ludkovsky

absolutely continuous component of µa relative to µ, the set of all partially admissible shifts we denote by M˜ µ . The proofs of Properties I-IV given below in §3.7 differ slightly from the proofs in Chapter II for Ks -valued measures. 3.7. Properties of Mµ and ρ from § 3.6. I. The set Mµ is a semigroup by addition, ρ(a + b, x) = ρ(a, x)ρ(b, x − a) for each a, b ∈ Mµ . II. If a ∈ Mµ , ρ(a, x) 6= 0 (mod µ), then µa ∼ µ, −a ∈ Mµ and ρ(−a, x) = 1/ρ(a, x − a). III. If ν ≪ µ and ν(dx)/µ(dx) = g(x), then Mµ ∩ Mν = Mµ ∩ [a : µ([x : g(x) = 0, g(x − a)ρµ (a, x) 6= 0]) = 0] and ρν (a, x) = [g(x − a)/g(x)]ρµ (a, x) (mod ν) for a ∈ Mµ ∩ Mν . IV. If ν ∼ µ, then Mν = Mµ . V. Let ν ≪ µ and dν(x)/dµ(x) = g(x), a ∈ Mµ only when a ∈ M˜ µ , ρν (a, x) = g(x − a)ρ˜ µ (a, x)/g(x) (mod ν).

(1)

Proof. Evidently Mµ ⊂ M˜ µ and for a ∈ Mµ functions ρµ and ρ˜ µ coincide. If a is not an admissible shift for µ, then it is not admissible for ν, hence Mν ⊂ M˜ µ . Then Z

X

f (x)νa (dx) = =

also

Z

f (x + a)ν(dx) = X

=

Z

X

Z

Z

X

Z

X

f (x)ρν (a, x)ν(dx)

f (x)ρν (a, x)g(x)µ(dx),

f (x + a)g(x)µ(dx) = X

˜ x)µ(dx) + f (x)g(x − a)ρ(a,

Z

X

Z

X

f (x)g(x − a)µa (dx)

f (x)g(x − a)λa (dx),

where λa ⊥ µ, consequently, ν ⊥ λa and νa ⊥ λa . If f = 0 (mod λa ), then from the equality Z

X

f (x)ρν (a, x)g(x)µ(dx) =

Z

X

˜ x)µ(dx) f (x)g(x − a)ρ(a,

that demonstrates Formula (1). VI. Let (X, B ) and (Y, E ) be two Banach spaces over K, µ : B → [0, ∞) be a bounded measure and T : X → Y be a K-linear mapping. Denote by ν(C) := µ(T −1C) the measure R on (Y, E ). If a ∈ Mµ , then b = Ta ∈ Mν and ρν (b, y) = X ρµ (a, x)µ(dx, B 0 |T −1 y), where µ(∗, B 0 |y) is the conditional distribution of the measure µ relative to the σ-algebra B 0 generated by sets T −1C, C ∈ E . In particular, if T is invertible, then ρν (b, y) = ρµ (T −1 b, T −1 y). Proof. This statement follows from Theorem 3.2.4, since each bounded non-zero measure µ induces the probability measure µ(A)/µ(X), A ∈ B . VII. Let X be a Banach space over K and on (X, B ) be given two non-negative bounded measures µ1 and µ2 ; µ = µ1 ∗ µ2 be the convolution of these measures. Then a ∈ Mµ for a ∈ Mµ1 , moreover, Z ρµ (a, x) =

X

ρµ1 (a, y)µ(dy, B 0 |U −1 x),

(1)

Real-Valued Measures

39

where µ(dy, B 0 |x) is the conditional measure corresponding to µ1 × µ2 relative to the σalgebra B 0 ⊂ B × B generated by all sets of the form {(x1 , x2 ) : x1 + x2 ∈ A}, a ∈ B , U : X × X → X so that U(x1 , x2 ) := x1 + x2 . Proof. This property also follows from Theorem 3.2.4 and also from Corollary 3.2.5. VIII. Let µ be a bounded non-negative measure on (X, B ). Define in Mµ the distance function Z r(a1 , a2 ) :=

X

|ρµ (a1 , x) − ρµ (a2 , x)|µ(dx).

(1)

Then (Mµ , r) is the complete pseudo-metric space. Proof. Suppose that an ∈ Mµ and limn→∞,m→∞ r(an , am ) = 0. We show that there exists a ∈ Mµ for which limn→∞ r(an , a) = 0. Mention that the sequence {an : n} is bounded in (X, k ∗ k). If kank k → ∞ with k → ∞, choose nk such that r(ank , ank+1 ) < 3−k . For B -measurable function with k f kL1 (µ) = 1 and with a bounded support we obtain Z

X

f (x + anN )µ(dx) ≥

Z

N−1 X

f (x + an )µ(dx) −

∑ r(an , an k

k+1

).

k=1

Choosing f so that X f (x + an1 )µ(dx) > 1/2 and taking the limit with N → ∞ we infer that R limN→∞ X f (x + anN )µ(dx) > 0, that is impossible for f with the bounded support, when kanN k → ∞. This means that {an : n} is bounded in X relative to the norm k∗k in the Banach space X. Therefore, we can consider without loss of generality, that {an : n} weakly converges to some a, since the topologically dual space denoted by X ∗ or by X ′ separates points in X for the locally compact field K (see also [NB85, Roo78] and the beginning of this chapter). From the relations R

Z

X

χe (z(x + an ))µ(dx) = χe (z(an ))

Z

X

χe (z(x))µ(dx) =

Z

X

χe (z(x))ρµ (an , x)µ(dx),

taking into account that limn→∞ z(an ) = z(a) for each z R∈ X ∗ . There exists the limit limn→∞ ρµ (an , x) = ρ(x) by the measure Rµ and limn→∞ X |ρµ (an , x) − ρ(x)|µ(dx) = 0 R we find that χe (z(a))R X χe (z(x))µ(dx) = X χe (z(x))ρ(x)µ(dx). From this we find that R ∗ X χe (z(x))µa (dx) = X χe (z(x))ρ(x)µ(dx) for each z ∈ X . Thus µa ≪ µ and ρ(x) = dµa (x)/dµ(x). 3.8. Definition and notes. A linear operator U on a Banach space X over K has a matrix representation u(i, j) = e˜i (Ue j ), where (e j : j) is a non-Archimedean orthonormal basis in X, e˜i are vectors of the topologically dual space X ∗ under the natural embedding (see §§ 2.2, 2.6). For a compact operator U − I we get that there exists limn→∞ det{ˆrn (U)} ∈ K which we adopt as the definition of det{U}, where rn : X → Kn is the projection operator with rˆn (U) := {u(i, j) : i, j = 1, . . . , n} (see also Appendix, use that limi+ j→∞ u(i, j) = 0). 3.9. Theorem. Let X be a Banach space over K, µ be a probability real-valued measure and T : X → X be a compact operator such that Re(1 − µˆ (z)) → 0 for |˜z(T 2 z)| → 0 and z ∈ X, z˜ ∈ X ′ corresponding to z. Then Mµ ⊂ (T X)∼ , where Y ∼ is a completion by k ∗ kY of a normed space Y . R Proof. For each a ∈ Mµ and b > 0 we have J(z) = R X [1 − [χe (˜z(x)) + χe (−˜z(x))]/2]ρ(a, x)µ(dx) ≤ J1 (z) + J2 (z), where J1 (z) := b X [1 − [χe (˜z(x)) +

40

Sergey V. Ludkovsky

χe (−˜z(x))]/2]µ(dx), J2 (z) = 2 (x: ρ(a,x)>b) µ(dx), J1 (z) → 0 for |˜z(T 2 z)| → 0, J2 (z) → 0 for b → +∞, consequently, lim|˜z(T 2 z)|→∞ Re(µa (X) − χe (˜z(a))ˆµ(z)) = 0. From 2 |1 − µˆ (z)| ≤ 2Re(1 − µˆ (z)) and µ(X) = µa (X) = 1 it follows that χe (˜z(a)) → 1 and z˜(a) → 0 for |˜z(T 2 z)| → 0. If |˜z(a)| < ε for |˜z(T 2 z)| < δ, then |˜z(a)|2 < ε2 |˜z(T 2 z)|/δ. Let (e j : j) be orthonormal basis in X such that T = (Ti, j : i, j ∈ N), limi+ j→∞ Ti, j = 0, we denote D j := supk |Tk, j |, hence |a j |2 < ε2 D2j /δ, a ∈ (T X)∼ , kakT X := sup j D j |a j | < ∞ (see Appendix). 3.10. Corollary. Let X and µ be the same as in § 3.9, then L ∩ Mµ is a set of the first category in X for each infinite-dimensional linear subspace L ⊂ X over K. Proof follows from the fact that (T B(X, 0, p j ))∼ =: A j is nowhere dense in L ∩ B(X, 0, p j ) for each j ∈ N, since A j is compact in X [Sch89]. The proof of Theorem 2 § 19[Sko74] can not be transferred on non-Archimedean X, there exists a compact V in X = c0 (ω0 , K) and its linear span Y = spanKV such that an ultranorm pE (x) := inf[|a| : x ∈ aE] (see Exer. 6.204 and 5.202[NB85]) produces from Y non-separable and non-Radonian Banach space l ∞ (ω0 , K) [DF91, Roo78], where E = co(V ) in Y . Note. Moreover, as it will be seen below from the proofs of Theorems 3.20 and 4.2 it follows the existence of a probability quasi-invariant measure µ : B f (X) → R with Mµ ⊃ H + GT , where GT is a compact subgroup in X with µ(GT ) > 0 such that µ(L) = 0 for real-valued µ and each linear subspace L in X with dimK L < ℵ0 (see Chapter II also respectively). 3.11. Definition. For a Banach space X over K and a measure µ : B f (X) → R, a ∈ X, kak = 1, a vector a is called an admissible direction, if a ∈ MµK := [z : kzkX = 1, λz ∈ Mµ and ρ(λz, x) 6= 0 (mod µ) (relative to x) and for each λ ∈ K] ⊂ X. Let a ∈ MµK we denote by L1 := Ka, X1 = X ⊖ L1 , µ1 and µ˜ 1 are the projections of µ onto L1 and X1 respectively, µ˜ = µ1 ⊗ µ˜ 1 be a measure on B f (X), given by the the following equation µ˜ (A ×C) = µ1 (A)˜µ1 (C) on B f (L1 ) × B f (X1 ) and extended on B f (X), where A ∈ B f (L1 ), C ∈ B f (X1 ). 3.12. Theorem. For a to be in MµK with a real-valued probability measure µ it is necessary and sufficient that the following conditions be satisfied (i-iv): (i) µ ≪ µ˜ ; (ii) µ1 ≪ m, where m is a real-valued Haar measure on L1 ; (iii) µ1 (dx)/µ(dx) := h(x) 6= 0 for |a(x)| > 0; (iv) µ([x : g(x − λa) 6= 0, g(x) = 0]) = 0 for g(x) = µ(dx)/˜µ(dx) whilst |λ| > 0. Proof. From µ˜ λa = µ1λa ⊗ µ˜ 1 for λ > 0 and µ1λa ≪ µ1 , hence λa ∈ Mµ˜ . In view of (iv) and and §§ 3.10 II, III the sufficiency follows, λa ∈ Mµ˜ ∩ Mµ . Now prove the necessity of Conditions (i − iv). At first we demonstrate that µ ≪ µ˜ . Recall the following proposition about measures equivalent with Haar measures. If G is a locally compact group and µ is a non-negative non-zero left invariant Haar measure on G, then a measure ν 6= 0 on G is left quasi-invariant if and only if it is equivalent with µ (see Proposition 11 §VII.1.9[Bou63-69]). From ρ(λz, x) 6= 0 (mod µ) it follows that µ1 ∼ m due to the cited just above proposition, where m is the Haar measure on L1 . For each A ∈ B for x ∈ L1 = aK denote by A1x the set of all y ∈ X 1 for which (x, y) ∈ A, where X = L1 ⊕ X1 , L1 ∩ X1 = {0}, X1 is the closed K-linear R

Real-Valued Measures

41

subspace in X. Then A1x ∈ B f (X1 ) and µ(A) = µ˜ (A) =

Z

X

Z

X

µ(x, A1x )µ1 (dx),

(1)

µ˜ 1 (A1x )µ1 (dx).

(2)

Therefore,R for proving µ ≪ µ˜ it is sufficient to show that µ(x, ∗) ≪ µ˜ 1 for each x. Since µ˜ 1 (B1 ) = X µ(x, B1 )µ1 (dx), then form µ˜ 1 (B1 ) = 0 it follows that µ(x, B1 ) = 0 for µ1 -almost all x. The density of µ1 relative to the Haar measure m on L1 at a point x for sufficiently large k(a, x)k is positive, where m is positive on a unit ball in L1 . Then there exist λ with sufficiently large |λ| for which µ(λa, B1 ) = 0. Using µ(λ1 a, ∗) ≪ µ(λa, ∗) for |λ1 | < |λ| we get that µ(λa, B1 ) = 0 for all λ. Hence from µ˜ (B1 ) = 0 it follows that µ(x, B1 ) = 0 for each x ∈ L1 , that is µ(x, ∗) ≪ µ˜ 1 . Calculate now dµ(x)/d µ˜ (x) = g(x). In view of Theorem 3.5.1 and Formulas (1, 2) we deduce that g(x) = dµ(PL1 x, PX1 x)/d µ˜ (PX1 x). Since µ(PL1 x − λa, ∗) ≪ µ(PL1 x, ∗) for λ 6= 0, then g(x − λa) = g(x)[dµ(PL1 x − λa, ∗)(PX1 x)/dµ(PL1 x, PX1 x)] (mod µ). (3) From this Condition (iv) follows. 3.13. Note. For Ks -valued measures Theorem 3.12 is untrue, since there are cases with µ that are not absolutely continuous relative to µ˜ . For example, for a probability measure µ = µ1 + µ2 6= 0, µ2 (X) = 0 with µ11 equivalent to the Haar measure m on L1 , µ12 = 0, when for µ2 all atoms (a j,2 ) are points and for each x ∈ L1 is accomplished ∑PL1 (a j,2 )=x µ2 (a j,2 ) = 0 and atoms (a j,1 ) of µ1 are such that PL1 (a j,1 ) 6= PL1 (al,2 ) for each j, l. We can choose (a j,s ) such that there exists a compact S ⊂ X with µ1 (S) = 0, µ2 (S) 6= 0, but µ˜ (S) = 0, since µ12 = 0. 3.14. Definition and notes. A measure µ : B f (X) → R for a Banach space X over K is called a quasi-invariant measure if Mµ contains a K-linear manifold Jµ dense in X. From § 3.7 and Definition 3.11 it follows that Jµ ⊂ MµK . Let (e j : j ∈ N) be an orthonormal basis in X, H = spanK (e j : j). We denote Ω := Ω(R) := [µ|µ is a real measure with a finite total variation on B = B f (X) and H ⊂ Jµ ]. A measure µ ∈ Ω(R) that can not be represented as a sum of two singular to each other measures from Ω(R) is called an extremal measure. B f (X)-measurable function h : X → R is called invariant for µ ∈ Ω(R), if h(a + x) = h(x) µ-almost everywhere by x for each a ∈ H. 3.14.1. Lemma. If ν and µ are measures from Ω and ν1 is an absolutely continuous component of ν relative to µ, then ν1 ∈ Ω and ν − ν1 ∈ Ω as soon as ν 6= ν1 . Proof. Put ν2 = ν − ν1 , ν2 ⊥ µ. If a ∈ H, then ν1a + ν2a ∼ ν1 + ν2 and ν2a ≪ ν1 + ν2 . Since ν2a ⊥ µa ∼ µ and ν1 ≪ µ, then ν2a ⊥ ν1 and ν2a ≪ ν, that is ν2 ∈ Ω. Then ν1a ≪ ν1 + ν2 and ν1a ⊥ ν2 , consequently, ν1a ≪ ν1 , ν1 ∈ Ω. 3.14.2. Corollary. If µ and ν are two extremal measures, then either µ ∼ ν or µ ⊥ ν. Proof. In the contrary case there would be µ = µ1 + µ2 , where µ1 ≪ ν, µ2 ⊥ ν, but this is impossible, if µ is the extremal measure. 3.14.3. Corollary. If µ is an extremal measure, then for each a ∈ X either µ ∼ µa or µ ⊥ µa . Proof. This follows from the fact that µa is the extremal measure as well.

42

Sergey V. Ludkovsky

3.14.4. Corollary. If µ is an extremal measure, ν ∈ Ω and ν ≪ µ, then ν is also extremal and ν ∼ µ. Proof. In the contrary case µ = µ1 + µ2 , where µ1 ≪ ν and µ2 ⊥ ν, µ1 , µ2 ∈ Ω. 3.14.5. Lemma. For a measure µ to be extremal it is necessary and sufficient for each invariant relative to µ function h(x) being a constant. Proof. Let h(x) be invariant for µ and there exists b such that µ({x : h(x) < b}) > 0 and µ({x : h(x) ≥ b}) > 0.R Put φ1 (x) = 0 for h(x) < b, while φ2 (x) = 0 for h(x) ≥ b, φ1 (x) + φ2 (x) = 1, µ j (A) = A φ j (x)µ(dx). Evidently, functions φ j (x) are also invariant for µ, j = 1, 2. Therefore, for a ∈ H we get Z

X

=

f (x)µaj (dx) Z

X

=

Z

j

f (x + a)µ (dx) = X

f (x + a)φ j (x + a)µ(dx) = =

Z

X

Z

X

Z

X

f (x + a)φ j (x)µ(dx)

f (x)φ j (x)ρµ (a, x)µ(dx)

f (x)ρµ (a, x)µ j (dx)

so that µa ≪ µ j and hence µ j ∈ Ω, also µ1 ⊥ µ. Since µ = µ1 + µ2 , then µ will not be the extremal measure. Let now µ = µ1 + µ2 , where µ j ∈ Ω, µ1 ⊥ µ2 . Denote φ j (x) = dµ j (x)/dµ(x). For each a ∈ H there are identities φ1 (x)φ2 (x + a) = φ1 (x)φ2 (x) = φ1 (x + a)φ2 (x) = 0 satisfied µalmost everywhere, since µ1 ⊥ µ2 , µ1a ⊥ µ2 , µ2a ⊥ µ1 and φ1 (x) + φ2 (x) = 1. Therefore, 0 = {(φ1 (x) − φ1 (x + a)) + (φ2 (x) − φ2 (x + a))}2 = (φ1 (x) − φ1 (x + a))2 + (φ2 (x) − φ2 (x + a))2 . Thus φ j (x) would be invariant different from constants functions for µ. 3.15. Denote by Ln the subspace spanK {e1 , . . . , en }, by Xn the orthogonal in the nonArchimedean sense complement to Ln , Xn = clX spanK {e j : j > n}, by B n and B n we denote σ-algebras generated by cylindrical subsets in Ln and Xn respectively. As usually µ(∗, U |x) denotes a conditional measure corresponding to µ relative to a σ-subalgebra U in B f (X) on the measure space (X, B f (X), µ). Lemma. If h(x) is a bounded invariant function for a measure µ, then h(x) = R h(y)µ(dy, B n |x) for all n ∈ N and µ-almost everywhere. X Proof. From § 2.37 the existence of a conditional measure follows. It is sufficient to ˜ demonstrate, that there exists a B n -measurable function h(x) µ-almost everywhere coinciding with h(x). Let φ (y) = C(b) exp(−|y|/b), where b > 0, C(b) > 0 is a constant so that b R Ln φb (y)mn (dy) = 1,R mn denotes the Haar measure on Ln with mn (B) = 1 for the unit ball B in Ln . Put hb (x) := Ln h(x + y)φb (y)mn (dy). From the B f (X)-measurability of h(x) it follows that hb (x) → h(x) for b → 0 by the measure µ due to Lemmas 2.8 and 2.30. Therefore, it is possible to choose such sequence bk that hbk (x) → h(x) µ-almost everywhere. MenR tion that the function hb (x + a) = Ln h(x + y)φb (y − a)mn (dy) is the continuous function by a ∈ Ln . But due to invariance of h we infer that hb (x + a) = hb (x) for µ-almost all x, consequently, hb (x + a) is constant as the function by a for µ-almost all x by the measure µ. This means that hb (x) = hb (Pn x) for µ-almost all x ∈ X, where Pn : X → Xn is the K-linear ˜ projection operator associated with the basis {e j : j}. The function h(x) = limk→∞ hbk (Pn x) is defined for all x, for which this limit exists. It is the desired function. T 3.15.1. Corollary. Let B µn be the completion of B n by the measure µ, B ∞ = n B µn ∩ B . If h is an invariant function for the measure µ, then h is B ∞ -measurable. j

Real-Valued Measures

43

Proof. In view of Lemma 3.15 the invariant function h(x) is B n -measurable, since such is the bounded invariant function arctan(h(x)). 3.15.2. Corollary. A function h(x) is invariant for a measure µ if and only if h(x) is B ∞ -measurable. Proof. The necessity follows from Corollary 3.15.1. The sufficiency follows from that for a B ∞ -measurable function h(x) for each n there exists a B n -measurable function hn (x) so that µ({x : h(x) = hn (x)}) = µ(X) > 0 and for each B n -measurable function hn (x) for all y ∈ Ln there is the equality hn (x + y) = hn (x). 3.15.3. Corollary. A probability measure is extremal if and only if µ takes only two values 0 and 1 on B ∞ . Proof. Only in this case all B ∞ -functions are equivalent to constants. 3.16. Theorem. A measure µ ∈ Ω(R) is an extremal measure if and only if µ is equivalent to ν ∈ R, where R := [ν ∈ Ω(R)| for each n there exists m > n such that ν(A ∩ B) = ν(A)ν(B) for each A ∈ Bn , B ∈ Bm ]. Proof. The necessity. Let µ be an extremal measure. Denote by µm n (∗|y) for all n ≤ m m m the conditional measure µ(∗, B n |y) on the σ-algebra calB and by µ the restriction of µ on B m . m n n We show that µm n (∗|y) ∼ µ for µ-almost all y. It is sufficient to show that µn (∗|y) ∼ µ m m for µ-almost all y, since µm n (∗|y) and µ are restrictions of these measures on B . n n Let λ (A|y) be defined for A ∈ B f (X), A ⊂ X , y ∈ Ln by the formula: λn (A|y) = µnn ((Pn )−1 A|Pn−1 y),

(1)

where Pn : X → Ln and Pn : X → Xn are the projection operator. We denote by λn the projection of µ on Xn and λn the projection of µ on Ln . It is sufficient to verify that for µn -almost all y measures λn (∗|y) and λn are equivalent. If consider X as Ln × Xn , then µ ∼ λn × λn . Evi1/pk

dently Ln ∈ B f (X), since Ln = ∞ , where Ab denotes the b-enlargement of a set A in k=1 Ln R X, b > 0. For each A ⊂ L with A ∈ B f (X) there is the equality µ(A×B) = A λn (dy)λn (B|y) Rn n and λn × λ (A × B) = A λn (dy)λn (B), then for proving the relation λn (∗|y) ∼ µn for µn m almost all y it remains to use Theorem 3.5.1. Thus µm n (∗|y) ∼ µ (mod µ). ∞ P ∞ Let now µn (∗|y) and µ ∞ be restrictions of these measures on B ∞ . Then µ∞ n (∗|y) ∼ µ ∞ ∞ (mod µ). In view of Corollary 3.15.3 the measure µ on B can take only two values 0 and ∞ ∞ ∞ 1. Then the measure µ∞ n (∗|y) being equivalent to it coincides with µ , that is µn (∗|y) = µ (mod µ). m ∞ If y ∈ X is such that µnn (∗|y) ∼ µn , then for all m ≥ n we get µm n (∗|y) ∼ µ and µn (∗|y) = m m µ∞ . Denote the set of such y by Cn . For y ∈ Cm we put ρm n (x|y) = dµn (∗|y)(x)/dµ (x). In accordance with Theorem 3.2.4 we infer that T

ρm+1 (x|y) = n

Z

X

m+1 ρm |x) n (z|y)µ(dz, B

(2)

(x, y), for each y ∈ Cn . Equality (2) implies that the sequence of functions gk (x) = ρN−k n k = 1, . . . , N − n forms the martingale, where N > n. In view of Note 2.45 we have that if A(β1 ,β2 ) with β2 > β1 ≥ 0 is the set of those x ∈ X intersects the interval (β1 , β2 ) for which the sequence ρm n (x, y) an infinite number of times T∞ for fixed n, x, y, then µ(A(β1 ,β2 ) ) = 0, since A(β1 ,β2 ) = k=1 Ak(β1 ,β2 ) , where Ak(β1 ,β2 ) is the

44

Sergey V. Ludkovsky

set of those x ∈ X for which the sequence ρm n (x, y) for fixed n, x, y intersects the interval (β1 , β2 ) not less than k times, while   Z m k µ(A(β1 ,β2 ) ) ≤ 2 ρn (x|y)dx /[β2 + (k − 1)(β2 − β1 )] X

= 2/[β1 + k(β2 − β1 )].

(3)

S∞

k,N The latter inequality follows from Ak(β1 ,β2 ) = N=n+1 Ak,N (β1 ,β2 ) , where A(β1 ,β2 ) is the set of those x ∈ X for which the sequence ρnn (x|y),. . . ,ρNn (x|y) intersects the interval (β1 , β2 ) not less than k times. Then ρNn (x|y), ρN−1 (x|y),. . . ,ρnn (x|y) is the martingale. Apply Note 2.45. n In accordance with Lemma 2.41 we infer that the sequence {ρm n (x|y) : m = n, n + 1, . . . }

is bounded (mod µ). From (2) and the Jensen’s inequality we deduce that Z

X

ψ(ρm+1 (x|y))µ(dx) ≤ n =

Z

X

Z

X

ψ(ρm n (x|y))µ(dx)

ψ(ρnn (x|y))µ(dx)

for each downward convex function ψ. Therefore the sequence {ρm n (x|y) : m} is uniformly integrable. For each B ∞ -measurable function h(x) we have the equalities: lim

Z

m→∞ X

h(x)ρm n (x|y)µ(dx) = lim

Z

m→∞ X

=

Z

X

h(x)µ∞ n (dx|y) =

Z

h(x)µm n (dx|y)

h(x)µ(dx) X

whenever these integrals exist, consequently, limm→∞ ρm n (x|y) = 1 (mod µ) for each y ∈ C0 . The function ρm (x|y) by y is B -measurable, as the function by x it is B n -measurable. n n m Therefore, there exists a B n × B n -measurable function ηm n (x|y) so that it is equal to ρn (x|y) m by (mod µ). Since ρn are uniformly integrable, then lim

Z

m→∞ X

n |ρm n (x|y) − 1|λ (dx) = lim

Z

m→∞ X

|ρm n (x|y) − 1|µ(dx) = 0.

(4)

m In accordance with Theorem 3.5.1 we can consider that ρm n (x|y) is B × B n . Integrating (4) by µ(dy) we infer that

lim

Z Z

m→∞ X

X

n |ρm n (x|y) − 1|λ (dx|y) = 0.

(5)

Consider the measure γn := λn × µn . From (5) we get that lim

Z Z

m→∞ X

X

|ρm n (x|y) − 1|γn (dx × dy) = 0.

(6)

From the definition we have also ρnn (x|x) = dµ(x)/dγn (x). Introduce further the conditional measures µ(∗, B n |y). If µn and µn (∗, B n |y) are restrictions of measures µ and

Real-Valued Measures

45

µ(∗, B n |y) on the σ-algebra B n , then µn (∗, B n |y) ∼ µn for µn -almost all y as for µm n (∗|y) and dµn (∗, B n |y)/dµn (x) = ρnn (x|x). l For each l < n < m denote by κm l,n the restriction of γn onto the σ-algebra B ∩ B m , m m while νl denotes the restriction of the measure µ onto the same σ-algebra. Put πl,n (x) = R n l m m dνm l (x)/dκl,n (x), then πl,n (x) = X ρn (z|z)γn (dz, B ∩ B m |x) due to Theorem 3.2.4. Applying Theorem 3.2.1 we infer that γn -almost everywhere there exists lim πm l,n (x) =

m→∞

Z

X

ρnn (z|z)γn (dz, B l |x) = dµl (x)/dγln (x),

(7)

where γln denotes the restriction of the measure γn onto the σ-algebra B l . m,k For all l < n < m < k define the σ-algebra B l,n consisting of all sets A ∩ B with A ∈ l m B ∩ B n , B ∈ B ∩ B k , we put m,k fl,n (x) =

Z

X

m,k −1 k l l [ρm n (z|z)] µ(dz, B l,n |x)πl,m (x)/[dµ (x)/dγm (x)].

(8)

From (6, 7) it follows that for all marked l and n lim lim f m,k (x) m→∞ k→∞ l,n

= 1 (mod µ).

(9)

We rewrite (8) in the form m,k fl,n (x) = [dγm (B l , ∗)(x)/dµ(B l , ∗)(x)][dµ(B l ∩ B k , ∗)(x)/dγm (B l ∩ B k , ∗)(x)]

(10)

m,k m,k ×[dγn (B l,n , ∗)(x)/dµ(B l,n , ∗)(x)],

where γm (U , ∗) and µ(U , ∗) denote the restrictions of the measures γm and µ onto a σR m,k (x)µ(dx) = 1. If n1 < n2 < · · · < n2N+1 is an algebra U , where U ⊂ B . Therefore, X fl,n n2k ,n2k+1 arbitrary sequence of numbers, then putting φk (x) = fn2k−2 ,n2k−1 (x) with n2k−2 = 0 for k = 1, B 0 = B , we get Z

N

∏ φk (x)µ(dx) =

X k=1

=

Z

X

Z

N

φk (x)µ(B 2 , dx) = · · · ∏ X k=2

φN (x)µ(B 2N−2 , dx) = 1.

(11)

Each function φk is µ-almost everywhere positive. From (9) it follows, that there exists a sequence {nk : k} such that µ-almost everywhere the infinite product ∏∞ k=1 φk (x) =: g(x) R converges. Introduce the measure ζ(A) := X g(x)µ(dx). From (11) we deduce that R X g(x)µ(dx) = 1, hence ζ is the finite measure. Moreover, µ ∼ ζ, since g(x) > 0 (mod µ). For a bounded B n2k−1 -measurable function ψ and a bounded B n2k−1 -measurable function ξ we get Z

X

ψ(x)ξ(x)ζ(dx) =

=

Z

k

X

Z

∞

X

ψ(x)ξ(x) ∏ φ j (x)µ(dx)

ψ(x) ∏ φ j (x)ξ(x) j=1

j=1

∞

∏

j=k+1

φ j (x)µ(dx).

46

Sergey V. Ludkovsky

The function ξ(x) ∏∞j=k+1 φ j (x) is B n2k -measurable, while ξ(x) ∏kj=1 φ j (x) is B n2k+1 measurable. Therefore, the function zk+1 (x) :=

Z

X

∞

∏

ξ(z)

φ j (z)µ(dz, B n2k+1 |x)

j=k+1

is B n2k ∩ B n2k+1 -measurable and Z

X

ψ(x)ξ(x)ζ(dx) =

Z

k

X

ψ(x) ∏ φ j (x)zk+1 (x)µ(B n2k+1 |dx).

(12)

j=1

n2k−2 |x). Since z Put zk−1 (x) := X ψ(z) ∏k−1 k−1 (x) is B n2k−1 -measurable, then j=1 φ j (z)µ(dz, B k−1 n 2k−2 z (x) is B ∩ B n2k−1 -measurable. Thus Equation (12) can be rewritten in the form:

R

Z

X

ψ(x)ξ(x)ζ(dx) = ×[dµ(B

Z

X

n2k−2

zk−1 (x)zk+1 (x)[dγn2k (B n2k−2 , ∗)(x)/dµ(B n2k−2 , ∗)(x)]

∩ B n2k+1 , ∗)(x)/dγn2k (B n2k−2 ∩ B n2k+1 , ∗)(x)]

n2k ,n2k+1 2k ,n2k+1 ×[dγn2k−1 (B nn2k−2 ,n2k−1 , ∗)(x)/dµ(B n2k−2 ,n2k−1 , ∗)(x)]µ(B n2k+1 , dx)

=

Z

z

k−1

X

=

(x)zk+1 (x)γn2k−1 (dx) = Z

k−1

X

ψ(x) ∏ φ j (x)µ(dx) j=1

Z Z

z

k−1

(x)µ(dx)

X

X

∞

ξ(x)

∏

Z

X

zk+1 (x)µ(dx)

φ j (x)µ(dx).

j=k+1

Particularly for ψ = 1 we get Z

X

ξ(x)ζ(dx) =

Z

ξ(x)

X

∞

∏

φ j (x)µ(dx),

j=k+1

for ξ = 1 from (12) we obtain Z

X

ψ(x)ζ(dx) =

Z

k−1

X

ψ(x) ∏ φ j (x)µ(dx). j=1

This means that Z

X

ψ(x)ξ(x)µ(dx) =

Z

X

ψ(x)ζ(dx)

Z

X

ξ(x)ζ(dx)/ζ(X).

(13)

The measure w(A) := ζ(A)/ζ(X) is equivalent with ζ, w ∼ ζ. In accordance with (13) for each pair of sets A ∈ B n2k−1 , B ∈ B n2k the equalities w(A ∩ B) = =

Z

are satisfied, hence w ∈ R.

X

χA (x)w(dx)

Z

Z

X

X

χA (x)χB (x)w(dx)

χB (x)w(dx) = w(A)w(B)

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47

The sufficiency. Suppose that µ ∈ R. We show that µ is the extremal measure. Let h(x) be a B ∞ -measurable bounded function. Then from the definition of the family R it follows that for each B n -measurable function ψ(x) the equality Z

X

ψ(x)h(x)µ(dx) =

Z

X

ψ(x)µ(dx)

Z

h(x)µ(dx)

(14)

X

is satisfied. The latter relation is satisfied for the set of functions ψ(x) closed relative to the bounded convergence. Thus (14) is satisfied for all bounded B -measurable funcR R R 2 2 = ( X h(x)µ(dx)) , consequently, X (h(x) − tions. Put in (14) ψ = h, then X h (x)µ(dx) R R 2 µ(dx) = 0, hence h(x) = h(x)µ(dx) (mod µ). Therefore, each B ∞ h(x)µ(dx)) X X measurable function coincides with a constant by the measure µ. Using Lemma 3.14.5 and Corollaries 3.15.1-3 we infer the sufficiency. 3.17. Theorem. For a Banach space X over K and each µ ∈ Ω(R) there exists [µy : R y ∈ Y ] ⊂ R and a measure ν on B∞ such that µ(A) = X µy (A)ν(dy) for each A ∈ B f (X), T ¯ n is a completion of Bn by a measure µ, ν = µ|B∞ , µy (A) is where B∞ := ( n B¯ n ∩ B f (X), B A f (X, µ)-measurable (see the notation in §§ 2.1, 3.14 and 3.16). Proof. We show that the measure µ ∈ Omega can be obtained by mixing of extremal measures so that there exists a family of measures µy ∈ R on (X, B ) for each y ∈ Y and a measure ν on (Y, E ) for which Z µ=

Y

µy ν(dy),

(1)

where µy (B) is E -measurable as the function by y for each marked subset B ∈ B . Take (Y, E ) = (X, B ∞ ) and ν = µ|B ∞ . We put µy (B) := µ(B, B ∞ |y), where µ(∗, B ∞ |y) is the conditional measure corresponding to µ relative to the σ-algebra B ∞ . From § 2.37 for the conditional measure it follows that µ(A) =

Z

µ(A, B ∞ |y)µ(dy) = X

Z

µy (A)ν(dy) X

for each A ∈ B . It remains to show that µy ∈ R for ν-almost all y. We first demonstrate that µy ∈ Ω. Take a ∈ Jµ and A ∈ B ∞ and a cylindrical (measurable) function f on X, then Z Z X

=

Z

X

f (x + a)µy (dx)ν(dy) = A

f (x)χA (x)ρµ (a, x)µ(dx) =

Z

X

Z Z X

A

f (x + a)χA (x)µ(dx) f (x)ρµ (a, x)µy (dx)ν(dy).

Hence for each such f and ν-almost all y we get: Z

y

f (x + a)µ (dx) = X

Z

X

f (x)ρµ (a, x)µy (dx).

(2)

Take a countable family C ′ of such cylindrical functions f for which (2) is satisfied and the closure of R-linear span of which is dense in the set of all cylindrical real-valued functions on X relative to the point-wise uniformly bounded convergence. Therefore, (2) is satisfied for all cylindrical functions for ν-almost all y, hence dµya (x)/dµy (x) = ρµ (a, x) (mod ν)

(3)

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Sergey V. Ludkovsky

also µya ∼ µy for each a ∈ Jµ and for ν-almost all y. Denote by m1 the non-negative Haar measure on K so that m1 (B(K, 0, 1)) = 1 and let m1 × ν be the measure on the product (K, B f (K) × (Y, E ), where B f (K) denotes the Borel σ-algebra on the field K. Then for m1 × ν-almost all (t, y) ∈ K × Y there is the relation y µta ∼ µy . Therefore, for ν-almost all y ∈ Y the set Sy ∈ B f (K) of those t ∈ K for which y µta ∼ µy for a marked y has m1 (K\Sy ) = 0. Moreover, Sy is the additive group. Suppose that / where Sy + h := {t + h : t ∈ Sy }, but m1 (Sy ∩ B(K, 0, r)) = h ∈ K \ Sy , then (Sy + h) ∩ Sy = 0, m1 ((Sy + h) ∩ B(K, 0, r)) for each |h| < r < ∞. Thus Sy = K for ν-almost all y. This demonstrates that for ν-almost all y ∈ Y each a ∈ Jµ is the admissible directional vector for the measure µy , that is by the definition ta ∈ Jµy for each t ∈ K. Thus µy ∈ Ω for ν-almost all y ∈ Y . We show next that µy ∈ R for ν-almost all y. Denote as usually µy (∗, B n |x) the conditional measure of µy relative to the σ-algebra B n for a marked y. Consider a cylindrical function f , a function gn (x) which is B n -measurable and bounded, also A ∈ B ∞ . For each B ∈ B ∞ we have µ(A ∩ B) =

Z

∞

µ(A, B |y)µ(dy) = B

Z

B

χA (y)µ(dy),

consequently, µy (A) = χA (y) (mod ν) for each A ∈ B ∞ . Then Z Z Z A X

=

Z Z

= Z Z X

=

f (x)µy (dz, B n |x)gn (x)µy (dx)ν(dy)

f (x)gn (x)µy (dx)ν(dy) =

A X

=

X

X

Z

A

gn (x)

Z

χA (x)gn (x)

Z Z

A X

gn (x)

Z

Z

A

f (x)gn (x)µ(dx)

f (z)µ(z, B n |x)µ(dx) X

Z

f (z)µ(z, B n |x)µy (dx)ν(dy) X

f (z)µ(z, B n |x)µy (dx)ν(dy).

(4)

X

From (4) we infer that Z

y

n

f (z)µ (dz, B |x) = X

Z

f (z)µ(dz, B n |x)

(5)

X

for ν-almost all y and each cylindrical function f . Since lim

Z

n→∞ X

n

f (z)µ(dz, B |x) =

Z

f (z)µ(dz) (mod µ), X

then for all f ∈ C ′ for all m for ν-almost all y we get Z

y

X

f (Pm x)µ (dx) = lim

Z

n→∞ X

f (Pm z)µy (dz, B n |x) (mod µy ).

(6)

Therefore, there exists E ∈ B ∞ with ν(E) = 1 so that for each y ∈ E Equality (6) is satisfied for each cylindrical function f .

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49

Suppose that h(x) is a bounded function invariant relative to the measure µy , y ∈ E. For each n the function h(x) is B n (y)-measurable, where B n (y) denotes the completion of B n by the measure µy , that is the minimal σ-algebra generated by B n and by all µy -null sets, µy ≥ 0. For each m < n we deduce that Z

X

f (Pm x)h(x)µy (dx) =

Z

h(x) X

Z

X

f (Pm z)µy (dz, B n |x)µy (dx).

Using (6) we find that Z

X

f (Pm x)h(x)µy (dx) = =

or that

Z

X

Z

Z

h(x) X

y

h(x)µ (dx) X

f (Pm x)[h(x) −

Z

Z

X

Z

X

f (Pm z)µy (dz)µy (dx)

f (Pm x)µy (dx)

h(z)µy (dz)]µy (dx) = 0. X

Taking into account the property of the family C ′ we come to the conclusion that h(x) = R y y y X h(z)µ (dz) (mod µ ). This means that each invariant function h(x) for the measure µ is y y µ -almost everywhere constant, consequently, µ ∈ R for each y ∈ E. 3.18. Theorem. If µ : B f (Y ) → R is a σ-finite measure on B f (Y ), Y is a complete separable ultrametrizable K-linear subspace such that co(S) is nowhere dense in Y for each compact S ⊂ Y , where K is an infinite non-discrete non-Archimedean field with a multiplicative ultranorm | ∗ |K . Then from Jµ = Y it follows that µ = 0. S Proof. Since µ is σ-finite, then there are (Y j : j ∈ H) ⊂ B f (Y ) such that Y = j∈H Y j and 0 < kµ|B f (Y j )k ≤ 1 for each j, where H ⊂ N, Y j ∩Yl = 0/ for each j 6= l. If card(H) = ℵ0 , then weRdefine a function f (x) = 1/[2 j |µ|(Y j )] for real-valued µ. Then we define a measure ν(A) = A f (x)µ(dx), A ∈ B f (Y ). Therefore, |ν|(Y ) ≤ 1 and Jν = Y , since f ∈ L1 (Y, µ, R). Hence it is sufficient to consider µ with kµk ≤ 1 and |µ|(Y ) = 1. For each n ∈ N in view of the Radonian property of Y there exists a compact Xn ⊂ Y such that |µ|(Y \ Xn ) < 1/n. In S Y there is a countable everywhere dense subset (x j : j ∈ N), hence Y = j∈N B(Y, x j , rl ) for each rl > 0, where B(Y, x, rl ) = [y ∈ Y : d(x, y) ≤ rl ], d is an ultrametric in Y , i.e. d(x, z) ≤ max(d(x, y), d(y, z)), d(x, z) = d(z, x), d(x, x) = 0, d(x, y) > 0 for x 6= y for each x, y, z ∈ Y . Therefore, for each rl = 1/l, l ∈ N there exists k(l) ∈ N such that |µ|(Xn,l ) > 1 − 2−l−n , Sk(l) T where Xn,l := j=1 B(Y, x j , rl ), consequently, |µ|(Y \ Xn ) ≤ 2−n for Xn := ∞ l=1 Xn,l . The subsets Xn are compact, since Xn are closed in Y and the metric d on Xn is completely bounded and Y is complete (see Theorems 3.1.2 and 4.3.29 [Eng86]). Then 0 < |µ|(X) ≤ 1 S for |µ|(Y \ X) = 0 for X := spanK ( ∞ n=1 Xn ). S The sets Y˜n = co(Yn ) are nowhere dense in Y for Yn = nl=1 Xl , consequently, spanKYn S / is dense in Y due to the Baire category are nowhere dense in Y . Moreover, (Y \ ∞ n=1 Yn ) 6= 0 theorem (see 3.9.3 and 4.3.26 [Eng86]). Therefore, y + X ⊂ Y \ X for y ∈ Y \ X and from Jµ = Y it follows that |µ|(X) = 0, since |µ|(y + X) = 0 (see §§ 2.38 and 3.14 above). Hence we get the contradiction, consequently, µ = 0. 3.19. Corollary. If Y is a Banach space or a complete countably-ultranormable infinitedimensional over K space, µ : B f (Y ) → R and Jµ = Y , then µ = 0.

50

Sergey V. Ludkovsky

Proof. The space Y is evidently complete and ultrametrizable, since its topology is given by a countable family of ultranorms. Moreover, co(S) is nowhere dense in Y for each compact S in Y , since co(S) = cl(Sbc ) is compact in Y and does not contain in itself any open subset in Y due to (5.7.5)[NB85] (for Y over R and real-valued measures see Theorem 4 § V.5.3[GV61]). 3.20. Theorem. Let X be a separable Banach space over a locally compact infinite field K with a nontrivial normalization such that either K ⊃ Qp or char(K) = p > 0. Then there are probability measures µ on X with values in R such that µ are quasi-invariant relative to a dense K-linear subspace Jµ . Proof. Let S( j, n) := p j B(K, 0, 1) \ p j+1 B(K, 0, 1) for j ∈ Z and j ≤ n, S(n, n) := pn B(K, 0, 1), w be the Haar measure on K considered as the additive group (see [HR79, Roo78]) with values in R. Then for each c > 0 and n ∈ N there are measures m on B f (K) such that m(dx) = f (x)v(dx), | f (x) |> 0 for each x ∈ K and | m(pn B(K, 0, 1)) − 1 |< c, m(K) = 1, | m | (E) ≤ 1 for each E ∈ B f (K) , where v = w, v(B(K, 0, 1)) = 1. Moreover, we can choose f such that a density ma (dx)/m(dx) =: d(m; a, x) be continuous by (a, x) ∈ K2 and for each c′ > 0, x and | a |≤ p−n : | d(m; a, x) − 1 |< c′ . For this we can define f for v = w, for example, to be at the beginning locally constant such that f (x) = a( j, n) for x ∈ S( j, n), where a( j, n) = rn( j−n) (1 − r−n )(1 − 1/p)p−n for j < n, a(n, n) = (1 − r−2n )p−n and m(E) := ∑{a( j, n)v(pn− j (E ∩ S( j, n))/v(pn− j S( j, n)) : j ∈ Z, j ≤ n}. Then we can take g(x) = f (x) + h(x) and y(dx) := g(x)v(dx), and a continuous h(x) : K → R, with sup{| h(x)/ f (x) |: x ∈ K} ≤ c” and 0 < c” ≤ 1/pn . More generally it is possible to take g ∈ L1 (K, B f (K), v, R) such that g(x) ≥ 0 for v-almost every x ∈ K and kgk = 1 and 1/2 1/2 ∏∞ n=1 (gn ∗ gn )(yn ) > 0 converges for each y = {yn : yn ∈ K, n ∈ N} in a proper dense subspace J in X = c0 , where gn (x) := g(x/an ), limn→∞ an = 0, 0 6= an ∈ K for each n ∈ N, 1/2 1/2 then use the Kakutani theorem 3.3.1, since H(µn , νn ) = (gn ∗ gn )(yn ) for the measure µn (dx) := gn (x)v(dx/an ) and its shifted measure νn (dx) := µn (−yn + dx). Let {m( j; dx)} be a family of measures on K with the corresponding sequence {k( j)} such that k( j) ≤ k( j + 1) for each j and limi→∞ k(i) = ∞, where m( j; dx) corresponds to the partition [S(i, k( j))]. The Banach space X is isomorphic with c0 (ω0 , K) [Roo78]. It has the orthonormal basis {e j : j = 1, 2, . . . } and the projectors Pj x = (x(1), . . . , x( j)) onto Kj , where x = x(1)e1 + x(2)e2 + · · · . Then there exists a cylindrical measure µ genj erated by a consistent family of measures y( j, B) = b( j, E) for B = P−1 j E and E ∈ B f (K ) [Bou63-69, DF91] where b( j, dz) = ⊗[m( j; dz(i)) : i = 1, . . . , j], z = (z(1), . . . , z( j)). Let L := L(t,t(1), . . . ,t(l); l) := {x : x ∈ X and | x(i) |≤ pa , a = −t −t(i) for i = 1, . . . , l, and a = −k( j) for j > l}, then L is compact in X, since X is Lindel¨of and L is sequentially compact [Eng86]. Therefore, for each c > 0 there exists L such that | µ(X \ L) |< c, since there is l ∈ N with | 1 − ∏[m( j; pk( j) B(K, 0, 1)) : j > l] |< c/2 (or due to the choice of a( j, n)). In view of the Prohorov theorem cited above (see also § IX.4.2[Bou63-69]) for real measures and due to Lemma 2.3 µ has the countably-additive extension on B f (X), consequently, also on the complete σ-field A f (X, µ) and µ is the Radon measure. Let z′ ∈ spanK {e j : j = 1, 2, . . . } and z” = {z( j) : z( j) = 0 for j ≤ l and z( j) ∈ S(n, n), j = 1, 2, . . . , n = k( j)}, l ∈ N, z = z′ + z”. In accordance with Lemma I.1.4 [Roo78] if K is a complete relative to its norm nonArchimedean field and f : B → K is a mapping such that f (0) = 0 and |[ f (x) − f (y)]/[x − y] − 1| ≤ c for each x, y ∈ B, where 0 < c < 1, B := B(K, 0, r) is the ball with center at zero

Real-Valued Measures

51

of radius r > 0, then f : B → B is the isometry. By Corollary 2.4 [Roo78] if (X, d) is a complete ultrametric space and if every decreasing sequence of values of the ultrametric d converges to zero, then (X, d) is spherically complete. In view of the Kakutani theorem cited above (see also [DF91]) and also two statements cited just above there are measures m( j; dz( j) such that ρµ (z, x) = ∏{d( j; z( j), x( j)) : j = 1, 2, . . . } = µz (dx)/µ(dx) ∈ L1 (X, µ, R) for each such z and x ∈ X, where d( j; ∗, ∗) = d(m( j; ∗), ∗, ∗) and µz (X) = µ(X) = 1. When char(K) = 0, then in this proof it is also possible to consider S( j, n) := π j B(K, 0, 1) \ π j+1 B(K, 0, 1), where π ∈ K, p−1 ≤ |π| < 1, π is the generator of the normalization group of K. 3.20.1. Theorem. Let X be a Banach space of separable type over a locally compact non-Archimedean field K and J be a dense proper K-linear subspace in X such that the embedding operator T : J ֒→ X is compact and nondegenerate, ker(T ) = {0}. Then a set M (X, J) of probability measures µ on B f (X) quasi-invariant relative to J is of cardinality 2c . If J ′ , J ′ ⊂ J, is also a dense K-linear subspace in X, then M (X, J ′ ) ⊃ M (X, J). Proof. Since X is of separable type over K, then we can choose for a given compact operator T an orthonormal base in X in which T is diagional and X is isomorphic with c0 over K such that in its standard base {e j : j ∈ N} the operator T has the form Te j = a j e j , 0 6= a j ∈ K for each j ∈ N, lim j→∞ a j = 0 (see Appendix). As in Theorem 3.20 take gn ∈ L1 (K, B f (K), v(dx/an ), R), gn (x) > 0 for v-a.e. x ∈ K and kgn k = 1 for each n, 1/2 1/2 for which converges ∏∞ n=1 (gn ∗ gn )(yn ), where the convolution is taken relative to the n measure v(dx/an ), for each y ∈ J and such that ∏m n=1 gn (xn )v(dxn /an ) =: νLn (dx ) satisfies n conditions of Lemma 2.3, where x := (x1 , . . . , xn ), x1 , . . . , xn ∈ K for each n ∈ N. Since gn are nonnegative, then it is sufficient to satisfy conditions of Lemma 2.3 for r = b > 0. The family of such sequences of functions {gn : n ∈ N} has the cardinality 2c , since in L1 c c 0 the subspace of step functions is dense and card(B f (X)) = ℵℵ 0 = c, card([0, ∞) ) = 2 , 2c×ℵ0 = 2c , where ℵ0 := card(N), c := card(R). The latter statement of this theorem is evident, since the family of all {gn : n} satisfying conditions above for J also satisfies such conditions for J ′ . 3.21. Note. For a given m = w new suitable measures may be constructed, if to use images of measures mg (E) = m(g−1 (E)) such that for a diffeomorphism g ∈ Di f f 1 (K) (see −1 § A.3) we have mg (dx)/m(dx) =| (g′ (g−1 (x)) |K , where | ∗ |K = modK (∗) is the modular function of the field K associated with the Haar measure on K, at the same time | ∗ |K is the multiplicative norm in K consistent with its uniformity [Wei73]. Indeed, for K and X = Kj with j ∈ N and the Haar measure v = w or v = w′ on X, vX := v with values either in R or in Ks for s 6= p and for a function f ∈ L1 (X, v, R) or f ∈ L(X, v, Ks ), respectively, due to Lemma 4 and Theorem 4 § I.2[Wei73], Theorems 9.2 and A.7[Sch84] we have: R R ′ (y)| v(dy), where mod (λ)v(dx) := v(λdx), λ ∈ K. For a f (g(y))|g f (x)v(dx) = K K A g(A) construction of new measures images of v may be used, for example, for analytic h on K (see 43.1 [Sch84]) with g = Ph, where P is the anti-differentiation such that g′ = h. Defining it at first locally we can then extend it on all K. Henceforward, quasi-invariant measure µ on B f (c0 (ω0 , K)) constructed with the help of projective limits or sequences of weak distributions of probability measures (µH(n) : n) are considered, for example, as in Theorem 3.20 such that

52

Sergey V. Ludkovsky

(i) µH(n) (dx) = fH(n) (x)vH(n) (dx), dimK H(n) = m(n) < ℵ0 for each n ∈ N, where S fH(n) ∈ L1 (H(n), vH(n) , R), H(n) ⊂ H(n + 1) ⊂ · · · , cl( n H(n) = c0 (ω0 , K), if it is not specified in another manner. Proposition 11 § VII.1.9 [Bou63-69] states, that if G is a locally compact group, G1 is its closed normal subgroup, G2 = G/G1 is the quotient group, π : G → G2 is the quotient mapping, u : G → G is a topological group automorphism of G such that u(G1 ) = G1 , u2 is an automorphism of G2 obtained by the factorization, then modG (u) = modG1 (u1 )modG2 (u2 ). For real-valued probability quasi-invariant measures for a sequence of weak distributions (µH(n) ) generated by µ in view of the proposition cited just above, Lemma 2.3 and Definition 3.14 this condition is satisfied, since cl(Jµ ) = c0 (ω0 , K). As will be seen below such measures µ are quasi-invariant relative to families of the cardinality c = card(R) of linear and non-linear transformations U : X → X. Moreover, for each V open in X we have µ(V ) > 0, when fH(n) (x) > 0 for each n ∈ N and x ∈ H(n). Let µ be a probability quasi-invariant measure satisfying (i) and (e j : j) be an orthonormal basis in Mµ , H(n) := spanK (e1 , . . . , en ), we denote by ˆ x) = lim ρn (Pn a, Pn x), ρˆ µ (a, x) = ρ(a, n→∞

ρn (Pn a, Pn x) := fH(n) (Pn (x − a))/ fH(n) (Pn x) ˆ x) = 0 in the contrary case, where for each a and x for which this limit exists and ρ(a, R ˆ x)µ(dx) = ˆ x), if ρ(a, Pn : X → H(n) are chosen consistent projectors. Let ρ(a, x) = ρ(a, ˆ x) 6= 1. If for some another basis (e˜ j : j) and ρ˜ is 1, ρ(a, x) is not defined when ρ(a, accomplished T ˜ x)]), then ρ(a, x) is called regularly depen(ii) µ(S) = 1, S := a∈Mµ [x : ρ(a, x) = ρ(a, dent from a basis. 3.22. Lemma. Let µ be a probability measure, µ : B f (X) → R, X be a Banach space over K, suppose that for each basis (e˜ j : j) in Mµ a quasi-invariance factor ρ˜ satisfies the following conditions: ˜ j , x), j = 1, . . . , N, are defined for a given x ∈ X and for each λ j ∈ K then a (1) if ρ(a ˜ ∑nj=1 λ j a j , x) is continuous by λ j , j = 1, . . . , N; function ρ( S (2) there exists an increasing sequence of subspaces H(n) ⊂ Mµ , cl( n H(n)) = ˜ n a, x) X, with projectors Pn : X → H(n), B ∈ B f (X), µ(B) = 0 such that limn→∞ ρ(P ˜ x) for each a ∈ Mµ and x ∈ = ρ(a, / B for which is defined ρ(a, x). Then ρ(a, x) depends regularly from the basis. S Proof. There exists a subset S dense in each H(n), hence µ(B′ ) = 0 for B′ = a∈S [x : ˜ x)]. From (1) it follows that ρ(a, ˜ x) = ρ(a, x) on each H(n) for x ∈ ρ(a, x) 6= ρ(a, / B′ . From ˜ x) for each a ∈ Mµ and x ∈ X \ (B′ ∪ B), spanK S ⊃ H(n) and (2) it follows that ρ(a, x) = ρ(a, consequently, Condition 3.21(ii) is satisfied. 3.23. Lemma. If a probability quasi-invariant measure µ : B f (X) → R satisfies Condition 3.21(i), then there exists a compact operator T : X → X such that Mµ ⊂ (T X)∼ , where X is the Banach space over K. Proof. This follows from Theorem 3.9 (see also Chapter II). 3.24. Let X be a Banach space over K, | ∗ |K = modK (∗), U : X → X be an invertible linear operator, µ : B f (X) → R be a probability quasi-invariant measure.

Real-Valued Measures

53

The uniform convergence of a (transfinite) sequence of functions on A f (V, ν)-compact subsets of a topological space V is called the Egorov condition, where ν is a measure on V . ˜ x)), where dom( f ) Theorem. Let pairs (x − Ux, x) and (x − U −1 , x) be in dom(ρ(a, ˜ − Ux, x) > 0, ρ(x ˜ − U −1 x, x) > 0 (mod µ). Then denotes a domain of a function f , ρ(x ν ∼ µ and ˜ −U −1 x, x), (i) ν(dx)/µ(dx) = |det(U)|K ρ(x if ρ depends regularly from the base, then ρ˜ may be substituted by ρ in formula (i), where ν(A) := µ(U −1 A) for each A ∈ B f (X). Proof. In view of Theorem 3.9 (and Lemma 3.23) there exists a compact operator T : X → X such that Mµ ⊂ (T X)∼ , consequently, (U − I) is a compact operator, where I is the identity operator. From the invertibility of U it follows that (U −1 − I) is also compact, ˜ moreover, there exists det(U) ∈ K. Let g be a continuous bounded function, g : H(n) → R, whence Z

X

φ(x)ν(dx) =

Z

˜ H(n)

g(x)[ fH(n) (U −1 x)/ fH(n) (x)]|det(Un )|K µH(n) (dx), ˜ ˜

˜ ˜ ˜ for φ(x) = g(P˜n x), where subspaces exist such that H(n) ⊂ X, (U −1 − I)H(n) ⊂ H(n), S ˜ ˜ ˜ ˜ + = X, Un := rˆn (U), rn = P˜n : X → H(n) (see §§ 3.8 and 3.21), H(n) ⊂ H(n cl( n H(n)) 1) ⊂ · R· · due to compactness of (UR− I). In view of the Fatou theorem Jm ≥ Jm,ρ , where ˜ − U −1 x, x)|det(U)|K µ(dx). Indeed, there Jm := X g(P˜m x)ν(dx) and Jm,ρ := X g(P˜m x)ρ(x exists n0 such that |u(i, j) − δi, j | ≤ 1/p for each i and j > n0 , consequently, |det(Un )|K = |det(U)|K for each n > n0 , where δi, j is the Kronecker delta-symbol. This implies that for each non-negative B f (X)-measurable function φ(x) the inequality Z

X

φ(x)ν(dx) ≥

Z

X

˜ −U −1 x, x)|det(U)|K µ(dx) φ(x)ρ(x

is satisfied, consequently, µ ≪ ν. Then due to the Fubini Theorem there exists lim [µH(n) (d P˜n x)/νH(n) (d P˜n x)] = µ(dx)/ν(dx) (mod ν). ˜ ˜

n→∞

Therefore, dµ(x)/dν(x) = 1/[ρ˜ µ (x −U −1 x, x)|det(U)|K ] (mod µ).

(1)

This means that the density of the absolute continuous component of ν relative to µ is: dν(x)/dµ(x) = ρ˜ µ (x −U −1 x, x)|det(U)|K .

(2)

Consider the measure λ obtained from µ by the mapping U −1 , then µ ≪ λ as soon as ρ˜ µ (x −Ux, x) > 0 (mod µ). Under the mapping U the measure µ transforms into ν, while λ into µ. Thus ν ≪ µ as well and inevitably µ ∼ ν. 3.25. Let X be a Banach space over K, | ∗ |K = modK (∗) with a probability quasiinvariant measure µ : B f (X) → R, also let U fulfils the following conditions: (i) U(x) and U −1 (x) ∈ C1 (X, X) (see § A.3); (ii) (U ′ (x) − I) is compact for each x ∈ X;

54

Sergey V. Ludkovsky (iii) (x −U −1 (x)) and (x −U(x)) ∈ Jµ for µ − a.e. x ∈ X; (iv) for µ-a.e. x pairs (x −U(x); x) and (x −U −1 (x); x)

are contained in a domain of ρ(z, x) such that ρ(x −U −1 (x), x) 6= 0, ρ(x −U(x), x) 6= 0 (mod µ); (v) µ(S′ ) = 1, where S′ := ([x : ρ(z, x) is defined and continuous by z ∈ L]) for each finite-dimensional L ⊂ Jµ ; (vi) there exists S with µ(S) = 0 and for each x ∈ X \ S and for each z for which there exists ρ(z, x) satisfying the following condition: limn→∞ ρ(Pn z, x) = ρ(z, x) and the convergence is uniform for each finite-dimensional L ⊂ Jµ by z in L ∩ [x ∈ Jµ : | x |≤ c], where c > 0, Pn : X → H(n) are projectors onto finitedimensional subspaces H(n) over K such that H(n) ⊂ H(n + 1) for each n ∈ N and cl ∪ {H(n) : n} = X; (vii) there exists n for which for all j > n and x ∈ X mappings V ( j, x) := x + Pj (U −1 (x) − x) and U( j, x) := x + Pj (U(x) − x) are invertible and lim j | detU ′ ( j, x) |=| detU ′ (x) |, lim j | detV ′ ( j, x) |= 1/ | detU ′ (x) |. Theorem. The measure ν(A) := µ(U −1 (A)) is equivalent to µ and (i) ν(dx)/µ(dx) =| detU ′ (U −1 (x)) |K ρ(x −U −1 (x), x). Proof. I. Let at first U be linear. In general, for a linear operator U with compact B = U − I there is the following decomposition U = SCDE, where Ct and E are upper triangular infinite matrices, D = diag{d( j) : j ∈ N}, operators C − I, D − I and E − I are compact in the corresponding orthonormal basis {e j : j} in X, S transposes a finite number of vectors in orthonormal basis (see § A.2). Moreover, there are det(C) = det(E) = 1, det(U) = det(D) 6= 0. II. Let Vn be a diagonal (or upper or lower triangular) operator on X such that (Vn − I)(X) = Pn L, where dimK L = k < ℵ0 , limn→∞ kVn − U1 k = 0, U1 is a diagonal (or lower or upper triangular) operator, there exists n0 such that ke j − Pn e j k ≤ 1/p for (e j ) in L, consequently, k ∑ j λ j Pn e j k = max j |λ j | for λ j ∈ K and dimK Pn L = k for n > n0 , in addition, limn→∞ supx∈L,kxk≤1 kx − Pn xk = 0. Then limn P˜n−1 (x − Vn−1 x) = x − U1−1 x, where P˜n x = Pn x ∈ Pn L for each x ∈ L. Due to Conditions (vi, vii) we get limn ρ(x − Vn−1 x, x) = ρ(x − U1−1 x, x) (mod µ). R From the Fatou theorem and § 3.21 it follows that, J ≥ J , where J = 1 1,ρ 1 X f (U1 x)µ(dx), R −1 J1,ρ := X f (x)ρ(x −U1 x, x)|detU1 |K µ(dx) for continuous bounded function f : X → [0, ∞). Analogously we proceed for the operator U1−1 instead of U1 . Using instead of f the function Φ(U1−1 x) := f (x)ρµ (x − U1−1 x, x) and Properties 3.7 we get that ρµ (U1 x − x,U1 x)ρµ (x − U1 x, x) = 1 (mod µ). Therefore, for U = U1U2 with diagonal U1 and upper triangular U2 and lower triangular U3 operators with finite-dimensional over K subspaces (U j − I)X, j = 1, 2, 3, the following equation is accomplished Z

f (Ux)µ(dx) = X

Z

X

f (x)ρµ (x −U −1 x, x)|detU|K µ(dx).

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55

If (S−1U − I)X = L, then from the decomposition given in (I) U = SU2U1U3 , we have (U j − I)X = L, j = 1, 2, 3 due to formulas from § A.1 (see Appendix), since corresponding non-major minors are equal to zero. (III). If U is an arbitrary linear operator satisfying the conditions of this theorem, then from (iv-vi) and (I, II) for each continuous Rbounded function f : X → [0, ∞) we have J ≥ Jρ, R where J := X f (U(x))µ(dx) and Jρ := X f (x)ρµ (x − U −1 (x), x)|detU|K µ(dx). Analogously for U −1 , moreover, ρ(x − U −1 (x), x)|detU|K =: h(x) ∈ L1 (X, µ, R), h(x) > 0 (mod µ), since there exists detU. (IV). Now let U be linear and (U − I)(X) = L, dimK L = k < ℵ0 , L ⊂ Jµ . Suppose U is polygonal, which means that there exists a partition X = ∪{Y (i) : i = 1, . . . , l}, U(x) = a(i) + V (i)x for x ∈ Y (i), where Y (i) are closed subsets, IntY (i) ∩ IntY ( j) = 0/ for each i 6= j, a(i) ∈ XR and V (i) are linear operators. Then U −1 is also polygonal, U ′ (x) = V ( j) for R x ∈ Y ( j) and X f (a(i) +V (i)x)µ(dx) = f (a(i) + x)ρµ (x −V −1 (i)x, x) ×|det(V (i))|K µ(dx) for each real Borel measurable function f and each i. From a( j) ∈ Mµ and § 3.7 we get R R f (x)ρ(x −V ( j)−1 (x − a( j)), x)|detV ( j)|K µ(dx). Let Hk, j := f (a( j) +V ( j)x)µ(dx) = X X [x ∈ X : V (k)−1 x = V ( j)−1 x], assume without loss of generality that V (k) 6= V ( j) or a(k) 6= a( j) for each k 6= j, since Y (k) 6= Y ( j) (otherwise they may be united). Therefore, Hk, j 6= X. If µ(Hk, j ) > 0, then from X ⊖ Hk, j ⊃ K it follows that Mµ ⊂ Hk, j , but cl(Hk, j ) = Hk, j and cl(Mµ ) = X. This contradiction means that µ(A) = 0, where A = [x : V (k)−1 (x − a(k)) = R R −1 V ( j) (x − a( j))]. Then X f (U(x))µ(dx) = X f (x)ρ(x −U −1 (x), x) |detU ′ (x)|−1 K µ(dx). (V). The field K is spherically complete. In view of the Hahn-Banach theorem for the Banach space X over K [NB85, Roo78] there are linear continuous functionals e˜ j such that there exists orthonormal basis (e j : j) in X with e˜i (e j ) = δi, j . Let s(i, j; x) := e˜i (U( j, x) − x) and s(i; x) = e˜i (U(x) − x) = lim j s(i, j; x) (lim is taken in C(X, K)), consequently, detU ′ ( j, x) = det(ds(i, j; x)e(k) : i, k = 1, . . . , j). Then for the construction of the sequence {U( j, ∗) : j} it is sufficient to construct a sequence of polygonal functions {a(i, j; x)}, that is a(i, j; x) = lk (i, j)(x) + ak for x ∈ Y (k), where lk (i, j) are linear functionS als, ak ∈ K, Y (k) are closed in X, Int(Y ( j)) ∩ Int(Y (k)) = 0/ for each k 6= j, m k=1 Y (k) = X, m < ℵ0 . For each c > 0 there exists Vc ⊂ X with µ(X \Vc ) < c, the functions s(i, j; x) and ¯ 1 s(i, j; ∗))(x, e(k),t) are equiuniformly continuous (by x ∈ Vc and by i, j, k ∈ N) on Vc . (Φ Choosing c = c(n) = 1/n and using δ-nets in Vc we get a sequence of polygonal mappings (Wn : n) converging by its matrix elements in L1 (X, µ, R), from Condition (i) follows that it may be chosen equicontinuous for matrix elements s(i, j; x), ds(i, j; x) and s(i, Pj x) by i, j (the same is true for U −1 ). Indeed, for Vc there is δ > 0 such that |s(i, j; x′ ) − s(i, j; x)| < c and ¯ 1 s(i, j; ∗))(x, e(k),t) − (Φ ¯ 1 s(i, j; ∗))(x′ , e(k),t ′ )| < c |(Φ (see also Theorem 2.11.I [Lud99t]) for each x, x′ ∈ Vc with |x−x′ | < δ, |t −t ′ | ≤ 1 and i, j, k ∈ N. Then we can choose in Vc a finite δ-net x1 , . . . , xr and define lq (i, j; x) = s(i, j; xq ) + (ds(i, j; xq ))(x − xq ) and apply the non-Archimedean variant of the Taylor theorem (see §29.4[Sch84] and Theorem A.5 in the Appendix). Then calculating integrals as above for Wn with functions f , using the Fatou theorem we get the inequalities analogous to written in (III) for J and Jρ of the general form. From

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Sergey V. Ludkovsky

ν(dx)/µ(dx) > 0 (mod µ) we get the statement of this theorem, since Z

f (A( j, x))µ(dx) = X

Z

X

f (x)ρµ (Pj (x − A(x)), x)|det(A( j, x)′ )|−1 K µ(dx),

where A = U, A( j, x) = U( j, x) or A = U −1 , A( j, x) = V ( j, x). 3.26. Note. For a linear invertible operator U Condition 3.25(i) is satisfied automatically, (ii) follows from (iii), hence from (iii) also follows (vii) (see the proof of Theorem 3.24). 3.27. Examples. Let X be a Banach space over the field K with the normalization group ΓK = ΓQp . We consider a diagonal compact operator T = diag(t j : j ∈ N) in a fixed orthonormal basis (e j ) in X such that kerT := T −1 0 = {0}. Let ν j (dx j ) = C(ξ j ) exp(−|(x j − q x0j )/ξ j | p )v(dx j ) for the Haar measure v : B f (K) → [0, ∞). We choose constant functions C(ξ j ) and C′ (ξ j ) such that ν j be a probability measure, where x0 = (x0j : j) ∈ X, x = (x j : R 0 0 j) ∈ X, x j ∈ K. Particularly, for q = 2 : K [(χe (x j − x j ) − χe (−x j + x j ))/(2i)]ν j (dx j ) = 0, since ν j is symmetric and [χRe (x) − χe (−x)] is the odd function, where χe is the same character as in § 2.6. Then K |x j − x0j |2 ν j (dx j ) = |ξ j |2 due to the formula of changing variables in § 3.21 and differentiation by the real parameter b of the following integral R I(b) = K exp(−|x|2p b2 )v(dx), then we take 1/|ξ| = b ∈ ΓK . Therefore, x0j and |ξ j |2 have in some respect a meaning looking like the mean value and the variance. N With the help of products ∞j=1 ν j (dx j ) as in § 3.20 we can construct a probability quasi-invariant measure µT on X with values in [0, 1] or Cs , since cl(T X) is compact in X T and spanK (e j : j) =: H ⊂ Jµ . From λ∈B(K,0,1)\0 cl(λT X) = {0} we may infer that for each T c > 0 there exists a compact Vc (λ) ⊂ X such that µ(X \ Vc (λ)) < c and λ6=0 Vc (λ) = {0}, R consequently, lim|λ|→0 X f (x)µλT (dx) = f (0) = δ0 ( f ), hence µλT is weakly converging to δ0 whilst |λ| → 0 for the space of bounded continuous functions f : X → R. q From Theorem 3.4 we conclude that from ∑∞j=1 |y j /ξ j | p < ∞ it follows y ∈ JµT . Then for a linear transformation U : X → X from ∑ j |e˜ j (x − U(x))/ξ j |2p < ∞ it follows that x − U(x) ∈ Jµ and a pair (x − U(x), x) ∈ dom(ρ(a, z)). Moreover, for ρ corresponding to µT conditions (v) and (vi) in § 3.25 are satisfied. Therefore, for such y and S ∈ A f (X, µ) a quantity |µ(ty + S) − µ(S)| is of order of smallness |t|q whilst t → 0, hence they are pseudodifferentiable of order b for 0 < Re(b) < 2 (see also § 4 below). Analogs of real-valued Wiener measures on spaces of functions from a nonArchimedean Banach space into C are given in [BV97, Sat94], that are quasi-invariant under some suitable choice. Another examples of measures with particular properties are given in § 1.6. 3.28. Theorem. Let A be a complete normed algebra over a locally compact infinite field K with a non-trivial non-Archimedean multiplicative norm. If a nontrivial real-valued measure µ on B f (A) is quasi-invariant relative to a dense subalgebra A′ (relative to linear shifts and left (or right) multiplication), then A is finite dimensional over K. Proof. If A is without a unity, then we can consider the algebra A1 with unity such that A ⊂ A1 and A1 is generated by A and K1. Therefore, µ can be extended to a measure µ′ on A1 quasi-invariant relative to A′ 1 , where A′ 1 is generated by A′ and K1, µ′ = µ × m, where m is the nontrivial Haar measure on K. Therefore, without loss of generality consider A with the unit element. In view of Lemma 3.23 there exists a compact operator T on A such that Mµ ⊂ (TA) . The measure µ is quasi-invariant relative to shifts x ∈ A′ ⊂ (TA) and relative

Real-Valued Measures

57

to left (or right) multiplication on (1 + y) ∈ A′ , where y ∈ A′ , then this implies that 1 is the compact operator on A (see also § 3.24). This is possible only when A is finite dimensional over K. 3.29. Theorem. Let A be a Banach space over a locally compact infinite field K supplied with a non-trivial non-Archimedean multiplicative norm. If µ is a non-trivial realvalued measure on B f (A) quasi-invariant relative to shifts from a dense K-linear subspace L′ in A, then there exists a nontrivial topological group G of K-linear automorphisms of A such that µ is also quasi-invariant relative to G. Proof. Take G consisting of K-linear operators satisfying conditions of Theorem 3.24. They form a multiplicative group, since the conditions of Theorem 3.24 are satisfied for the product UV of two operators U and V satisfying these conditions, also det(UV ) = det(U)det(V ) for each pair of operators in G and ρ˜ satisfies the co-cycle condition. In the topology of G inherited from the Banach space L(A) of bounded K-linear operators this group G is the topological group.

1.4.

Pseudo-differentiable Measures

4.1. Definition and notes. A function f : K → R is called pseudo-differentiable of order b, if there exists the following integral: PD(b, f (x)) :=

Z

[( f (x) − f (y)) × g(x, y, b)]dv(y).

(1)

K

We introduce the following notation PDc (b, f (x)) for such integral by B(K, 0, 1) instead of the entire K. Where g(x, y, b) :=| x − y |−1−b with the corresponding Haar measure v with values in R, where b ∈ C and |x|K = p−ord p (x) . Obviously, the definitions of differentiability of measures can not be transferred from [BS90, DF91] onto the case considered here. This is the reason why the notion of pseudodifferentiability is introduced here. A quasi-invariant measure µ on X is called pseudodifferentiable for b ∈ C, if there exists PD(b, g(x)) for g(x) := µ(−xz + S) for each S ∈ A f (X, µ) with |m|(S) < ∞ and each z ∈ Jµb , where Jµb is a K-linear subspace dense in X. For a fixed z ∈ X such measure is called pseudo-differentiable along z. For a one-parameter subfamily of operators B(K, 0, 1) ∋ t → Ut : X → X quasi-invariant measure µ is called pseudo-differentiable for b ∈ C, if for each S the same as above there exists PDc (b, g(t)) for a function g(t) := µ(Ut−1 (S), where X may be also a topological group G with a measure quasi-invariant relative to a dense subgroup G′ (see [Lud99t, Lud98s, Lud00a]). 4.2. Let µ, X, and ρ be the same as in Theorem 3.24 and F be a non-Archimedean Fourier transform defined in [VVZ94, Roo78]. Theorems. (1) g(t) := ρ(z + tw, x) j(t) ∈ L1 (v, K → C) := V , moreover, there exists F(g) ∈ C0 (K, C) and lim|t|→∞ F(g)(t) = 0 for µ and v with values in R, where z and w ∈ Jµ , t ∈ K, j(t) is the characteristic function of a compact subset W ⊂ K. In general, may be k(t) := ρ(z + tw, x) ∈ / V. (2) Let g(t) = ρ(z + tw, x) j(t) with clopen subsets W in K. Then there are µ, for which there exists PD(b, g(t)) for each b ∈ C. If g(t) = ρ(z + tw, x), then there are probability measures µ, for which there exists PD(b, g(t)) for each b with 0 < Re(b) or b = 0.

58

Sergey V. Ludkovsky

(3) Let S ∈ A f (X, y), |µ|(S) < ∞, then for each b ∈ U := {b′ : Re b′ > 0 or b′ = 0} there is a pseudo-differentiable quasi-invariant measure µ . Proof. We consider the following additive compact subgroup GT := {x ∈ X|kx( j)k ≤ pk( j) for each j ∈ N} in X, where T = diag{d( j) ∈ K : |d( j)| = p−k( j) for each j ∈ N} is a compact diagonal operator. Then µ from Theorem 3.20 is quasi-invariant relative to the following additive subgroup ST := GT + H, where H := spanK {e( j) : j ∈ N}. Moreover, for each z and w ∈ GT and R > 0 there is c > 0 such that ρ(z + uw, x) = ρ(z + sw, x) for |u − s| < c and x ∈ B(X, 0, R), if all functions f j in the proof of Theorem 3.20 are locally constant (that is, f are defined on Ke j ⊂ X). In general, for each b ∈ C we can choose a sequence h j (x) with ∑∞j=0 supx∈K (|h j (x)/(h j (x) + f j (x))|r j (x)) < c′ for suitable fixed c′ > 0 and functions r j : K → [0, ∞) with lim|x|→0 r j (x) = ∞. Carrying out calculations and using the fact that µ is the probability quasi-invariant measure we get the pseudo-differentiability of µ. Using the Riemann-Lebesgue theorem (see it in [VVZ94]) we get the statement of this theorem for F(g). 4.2.1. Theorem. Let X be a Banach space of separable type over a locally compact non-Archimedean field K and J be a dense proper K-linear subspace in X such that the embedding operator T : J ֒→ X is compact and nondegenerate, ker(T ) = {0}, b ∈ C. Then a set P b (X, J) of probability measures µ on B f (X) quasi-invariant and pseudo-differentiable of order b relative to J is of cardinality 2c . If J ′ , J ′ ⊂ J, is also a dense K-linear subspace in X, then P b (X, J ′ ) ⊃ P b (X, J). Proof. As in § 3.20.1 choose for T an orthonormal base in X in which T is diagonal and X is isomorphic with c0 over K such that in its standard base {e j : j ∈ N} the operator T has the form Te j = a j e j , 0 6= a j ∈ K for each j ∈ N, lim j→∞ a j = 0 (see Appendix). Take gn from § 3.20.1, where gn ∈ L1 (K, B f (K), v(dx/an ), R), satisfy conditions there and 1 such that there exists limm→∞ PD(b, ∏m n=1 gn (xz)) ∈ L (X, B f (X), ν, C) by the variable x for each z ∈ J, where x ∈ K. Evidently, P (X, J) ⊂ M (X, J). The family of such sequences of functions {gn : n ∈ N} has the cardinality 2c , since in L1 (ν) the subspace of step functions is dense and the condition of pseudo-differentiability is the integral convergence condition (see §§ 4.1 and 4.2). The latter statement of this theorem is evident, since the family of all {gn : n} satisfying conditions above for J also satisfies such conditions for J ′ . 4.3. Let X be a Banach space over K, b0 ∈ R ∪ {+∞} and suppose that the following conditions are satisfied: (1) T : X → X is a compact operator with ker(T ) = {0}; (2) a mapping F˜ from B(K, 0, 1) to CT (X) := {U : U ∈ C1 (X, X) and (U ′ (x) − I) is a compact operator for each x ∈ X, there is U −1 satisfying the same conditions as U} is given; ¯ 1Ut (x + h, x) are continuous by t, that is, F˜ ∈ C1 (B(K, 0, 1), ˜ = Ut (x) and Φ (3) F(t) CT (X)); (4) there is c > 0 such that kUt (x) −Us (x)k ≤ kT xk for each x ∈ X and |t − s| < c; (5) for each R > 0 there is a finite-dimensional over K subspace H ⊂ X and c′ > 0 such that kUt (x) −Us (x)k ≤ kT xk/R for each x ∈ X ⊖ H and |t − s| < c′ with (3 − 5) satisfying also for Ut−1 . Theorem. On X there are probability quasi-invariant measures µ which are pseudodifferentiable for each b ∈ C with R ∋ Re(b) ≤ b0 relative to a family Ut . Proof. From Conditions (2,3) it follows that there is c > 0 such that |det(Ut′ (x))| = |det(Us′ (x))| for µ-a.e. x ∈ X and all |t − s| < c, where quasi-invariant and pseudo-

Real-Valued Measures

59

differentiable measures µ on X relative to ST may be constructed as in the proof of Theorems 3.20 and 4.2. From (1 − 5) it follows that conditions of Theorem 3.25 are satisfied for each Ut . From (3, 5) it follows that for each R > 0 and b > 0 there exists c′ > 0 such that |ρµ (x −Ut−1 (x), x) − ρµ (x −Ut−1 (x), x)| ≤ b for each |t − s| < c′ and x ∈ B(X, 0, R). If during construction of µ to use only locally constant f j (x j ) with h j = 0, then we can take ∞ ≥ ˜ = µ(E S) ˜ b0 ≥ 0. Indeed, let T = SCDE be a decomposition from the appendix and µA (S) −1 −1 −1 with A = E (or C), S˜ ∈ B f (X), then ρµA (x − Ut (x), x) = ρµ (Ex − Ut (Ex), Ex) and c(x) := kUt−1 (Ex) − Us−1 (Ex)k ≤ kT Exk for x ∈ X and c(x) ≤ kT Exk/R for x ∈ X ⊖ H (for each R > 0 there is H ⊂ X with dimK H < ∞), due to compactness of E − I and C − I we can choose H such that A−1 H = H, AH = H. We use also images of quasiinvariant measure constructed with the help of compact diagonal operator (D′ − I), where |D j − 1| < |D′j − 1|pk( j) , lim j k( j) = −∞. Then sup{kUt (x) − Ut (0)k : x ∈ B(X, 0, R)} ¯ 1Ut kB(X,0,R) × kxk. From the fact that H is finite-dimensional and the existence of ≤ kΦ the orthogonal projector (that is, corresponding to the decomposition into the direct sum) πH : X → H it follows that kUta (e j ) − Usa (e j )k ≤ b j for each j ∈ N and each |t − s| < c′ , where lim j→∞ b j = 0, a = 1 or a = −1. This guarantees pseudo-differentiability of µ. 4.4. Let X be a Banach space of separable type over K, µ be a probability quasi-invariant measure µ : B f (X) → R, that is pseudo-differentiable for a given r with Re(r) > 0, Cb (X) be a space of continuous bounded functions f : X → R with k f k := supx∈X | f (x)|. Theorem. For each a ∈ Jµ and f ∈ Cb (X) is defined the following integral:  Z Z f (x)[µ(−λa + dx) − µ(dx)] g(λ, 0, r)v(dλ) (i) l( f ) = K

X

and there exists a measure ν : B f (X) → C with a bounded variation. For 0 < r ∈ R this ν is a mapping from B f (X) into R such that (ii) l( f ) =

Z

f (x)ν(dx), X

where v is the Haar measure on K with values in [0, ∞), moreover, ν is independent from f and may be dependent on a ∈ Jµ . We denote ν =: D˜ ra µ. Proof. From Definition 4.1 and the Fubini and Lebesgue theorems it follows that there exists  Z Z ( f (x + λa) − f (x))g(λ, 0, r)µ(dx) v(dλ) = l( f ), lim j→∞ K\B(K,0,p− j )

X

that is (i) exists. Let (iii) l j (V, f ) :=

Z

K\B(K,0,p− j )

Z

V

 f (x)(µ(−λa + dx) − µ(dx))g(λ, 0, r) v(dλ),

where V ∈ B f (X). Consider the measure νλ,a := (µλa − µ)g(λ, 0, r) for a ∈ Jµ , λ ∈ K. Let B be some subset in X and Bh be an h-enlargement of B so that Bh := {x ∈ X : infy∈B kx − yk ≤ h}, where h > 0. We show that for each b > 0 there exists a compact C such that |νλ,a |(X \C) < b for each λ 6= 0, where |ν| denotes the variation of a real-valued

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Sergey V. Ludkovsky

measure ν. For demonstrating this it is sufficient to prove that for each b > 0 and h > 0 there exists a compact C1 so that |νλ,a |(X \ C1h ) < b for each λ 6= 0. Indeed, if the latter is satisfied, then choose hn > 0 monotonously decreasing and converging to zero with n tending to the infinity and take a compact Cn so that |νλ,a |(X \Cnhn ) < b/2n for each λ 6= 0. T hn Then C = ∞ n=1 Cn is compact, since it is closed and C has a finite hn -net for each n. ∞ hn n Moreover, |νλ,a |(X \C) ≤ ∑∞ n=1 |νλ,a |(X \Cn ) ≤ ∑n=1 b/2 = b for each λ ∈ K \ {0}. Suppose now the contrary that for some b > 0 and h > 0 there does not exist any compact C so that |νλ,a |(X \Ch ) < b for all λ ∈ K\{0}. The field K is locally compact and separable, the Banach space X is of separable type over K, hence X is the Radon space, since X is the separable complete metric space. Therefore, each measure |νλ,a | is Radon for λ 6= 0. Choose a sequence of compacts Cn and λn 6= 0 with |νλn ,a |(X \ Cn ) > b/2, |νλn ,a |(X \ Sn Sn−1 h j=1 C j ) ≤ b/8 with Cn ⊂ X \ j=1 C j for each n ∈ N. The norm in X is non-Archimedean, hence the sets Cnh are pairwise disjoint for different n, since Cn ×C j is compact and a continuous real-valued function on a compact achieves at some point in it its infimum, so that infy∈Cn ,x∈C j kx − yk > h for each 1 ≤ j < n. For each n there exists a continuous function fn (x) equal to zero on X \Cnh and such that R X f n (x)νλn ,a (dx) ≥ b/2 and supx∈X | f n (x)| ≤ 1. For each subsequence {nk : k} the series ∑k fnk (x) converges and is the continuous function with the norm k f kC0 := supx∈X | f (x)| ≤ 1. b For each prime number s > 1 we construct a function gs (x) := ∑ j fsn j (x), where a finite or infinite set {n j : j} is chosen so that for each prime number s > 1 the inequality Z ∑ fsn j (x)νλ m ,a (dx) > b/8 (1) s X j

for some m > nk is satisfied. We describe the inductive procedure of choice to achieve this. Take n1 = 1 and if n1 < n2 < · · · < nk are chosen, then put Z  k    nk+1 := inf m : m > nk , and ∑ fsn j (x) + f pm (x) νλsm ,a (dx) ≥ b/4 . X

(2)

j=1

If the set in the curled brackets is is void, then {n1 , . . . , nk } is the desired set of indices. In accordance with the definition of nk+1 for nk < m < nk+1 the inequality Z  k  X ∑ fsn j (x) + f pm (x) νλsm ,a (dx) < b/4 j=1

is fulfilled. Therefore,

Z X

k

∑

j=1

fs (x)νλsm ,a (dx) ≥ b/4 nj

(3)

for each m such that either nk < m ≤ nk+1 when nk+1 exists or nk < m if nk+1 does not exist. Thus Z ∑ fsn j (x)νλ m ,a (dx) s X j

Z ≥ X

∑

Real-Valued Measures Z fsn j (x)νλsm ,a (dx) − ∑ fsn j (x)νλsm ,a (dx)

61

X n j >m

n j ≤m

sm   [ ≥ b/4 − |νλsm ,a | X \ Cl ≥ b/8 l=1

and the desired set {nk : k} is constructed together with the function gs (x). We have that lim|λ|→∞ |νλ,a |(X) = 0, since r > 0, |µ|(X) < ∞, |µλa |(X) = |µ|(X), while lim |λ| → ∞g(λ, 0, r) = 0 for r > 0. Therefore, due to |νλn ,a |(X) ≥ b, where b > 0, we can without loss of generality take a bounded sequence of λn . Then due to local compactness of the field K this sequence has an accumulation point. So we can consider the case when the sequence λn converges with n tending to the infinity. Since lim

Z

n→∞ X

gs (x)νλn ,a (dx) = lim

then

Choose 0 < bs ≤ 1 such that

n→∞ X

Z

n→∞ X

then n→∞

gs (x)νλsm ,a (dx),

Z lim gs (x)νλ ,a (dx) ≥ b/8. n n→∞ X lim

lim

Z

Z

bs gs (x)νλn ,a (dx) ≥ b/8,

N

∑ bs gs (x)νλ ,a (dx) ≥ k(N)b/8, X n

s=2

where k(N) denotes the number of prime numbers in [2, N]. But k ∑Ns=2 bs gs (x)kC0 ≤ 1 b for each N, since kgs kC0 ≤ 1 for each s, while for different s the functions gs can not be b sumultaneously non-zero. The functional l( f ) is the limit of the convergent sequence of C-linear and continuous functionals on the space of continuous bounded complex-valued functions f : X → C, hence l( f ) is also C-linear and continuous. This means that there exists a constant y > 0 with |l( f )| ≤ yk f kC0 for each f ∈ Cb0 (X, C). This implies that b

k(N)b/8 ≤ lim

n→∞

Z

N

∑ bs gs (x)νλ ,a (dx) ≤ y, X n

(4)

s=2

but this is impossible for k(N) > 8y/b, since limN→∞ k(N) = ∞. This contradiction demonstrates that the claimed above compact set exists. Hence for each c > 0 there exists a compact Vc ⊂ X with |νλ |(X \Vc ) < c for each |λ| > 0, R where νλ (A) := K\B(K,0,|λ|) [µ(−λ′ a+A)−µ(A)]g(λ′ , 0, b)v(dλ′ ) for A ∈ B f (X). Also there are δ > 0 and Vc such that −λ′ a +Vc ⊂ Vc and k[X \Vc ) △ (−λ′ a + (X \Vc ))]kµ = 0 for each |λ′ | < δ (see also Theorem 7.22 [Roo78]), where A △ B := (A \ B) ∪ (B \ A). Each continuous linear functional on Cb (V ) for compact V has the form (ii) (see [Bou63-69], A.5 [Sch84]), that is, for l j (V, f ) there exists a measure νλ (c, ∗). In view

62

Sergey V. Ludkovsky

of the Alaoglu-Bourbaki theorem each bounded subset W in [Cb (Vc )]′ is precompact (i.e., cl(W ) is compact) in the weak topology, where Cb (Vc ) is separable. For each c > 0 we have l( f ) =

Z

f (x)ν(c, dx) + O(c × sup | f (x)|)

Vc

x∈V / c

such that ν(c, ∗) are measures on B f (Vc ). Choosing Vc ⊂ Vc′ for c > c′ and defining ν(A) := limc→+0 ν(c, A ∩ Vc ) for each A ∈ B f (X) in view of the Radon-Nikodym theorem about convergence of measures (see [Con84]) we get (i) with the help of the theorem about extension of measures, since the minimal σ-field σBco(X) generated by Bco(X) coincides with B f (X). 4.5. Theorem. If µ is probability real-valued quasi-invariant measure on a Banach space X over K = Qp , a ∈ Jµ and µ is pseudo-differentiable of order 1, then for each f ∈ L∞ (X, µ, R) with supp( f ) ⊂ B(X, 0, R), ∞ > R = R( f ) > 0 the following equality is accomplished: (i)

Z

X

λ=β ˜ 1 [D−1 λ f (x + λa)]|λ=0 (Da µ)(dx)

=

Z

X

[ f (x + βa) − f (x)]µ(dx),

moreover, pseudo-differential D˜ 1a µ := ν in the direction a from Theorem 4.4 given by formula (ii) characterizes µ uniquely. Proof. From the restrictions on f it follows that f (x + λa) =: ψ(λ) ∈ E′ , where E′ is the topologically dual space to the space E of basic locally constant functions [VVZ94]. By formula (1.5) § 9[VVZ94] for them there is the following equality Da Db ψ = Db Da ψ = Da+b ψ for a + b 6= −1, a, b ∈ R ( Daλ denotes a pseudo-differentiation of functions by λ). In view of Theorem in § 6.3[VVZ94] E′ = [φ ∈ D′ : supp(φ) is compact ], where D = [φ ∈ E : β −1 supp(φ) is compact ]. Since [ξ]|0 := ξ(β) − ξ(0) for ξ(λ) = D−1 λ ψ(λ) and (D Dψ)(λ) = ψ(λ), hence due to the definition of D−1 λ we have: p2 (p + 1)−1

Z

β

X

˜1 [D−1 λ f (x + λa)]|0 (Da µ)(dx) = lim

−1−α −1 2

(1 − p

)

Z

Z

α→−1 Qp Qp

−1

p (1 + p)

Z

X

(1 − pα )

(µ(−a(λ′ + λ”) + dx) − µ(−aλ′ + dx))

 ′ −1−α |λ”|−2 |λ − λ | µ(dx) v(dλ′ )v(dλ”). p p In view of the Fubini and Lebesgue theorems we establish Formula (i). Taking all different f ∈ L∞ (X, µ, R) such that ψ(λ) = f (x + aλ) are locally polynomial by λ and using the nonArchimedean variant of the Stone-Weiestrass theorem by Kaplanski [Sch84] we get that µ is characterized by the left side of (i) uniquely. 4.6. (Properties of D˜ 1a µ for real-valued measures µ (see also §§ 4.4. and 4.5), where K = Qp .) Proposition.1. If µ is pseudo-differentiable in a direction a ∈ Jµ1 , a 6= 0, then for each d ∈ Jµ1 a measure µd (A) := µ(A − d), A ∈ B f (X), is also pseudo-differentiable in a direction a, moreover, D˜ 1a (µd ) = (D˜ 1a µ)d . If a ∈ Jµb , then D˜ bλa µ exists for each λ ∈ K.

Real-Valued Measures

63

Proof. From §§ 4.4 and 4.5 it follows that Z

d f (x)(D˜ −1 a µ) (dx) =

X

=

Z Z

K X

Z

X

d [ f (x + λa) − f (x)]|λ|−2 K µ (dx)v(dλ)

f (x)(D˜ 1a (µd ))(dx).

The last statement follows from Definition 4.1 with the help of § 3.21. 1 Proposition.2. If a ∈ Jµ1 , ν = D˜ 1a µ and D−1 λ ρν (λa, x) =: φ(x) ∈ L (X, ν, R) by x, then λa µ ≪ ν. Proof. In view of Formula 4.5.(i) the following equality is accomplished: Z

λa

f (x)µ (dx) = X

Z

f (x)µ(dx) + X

Z

ξ=λ

X

f (x)[D−1 ρν (λa, x)]|ξ=0 ν(dx). ξ

Taking f (x) = χA (x), where χA is a characteristic function of A ∈ B f (X) we get µλa (A) = µ(A) for every |λ| > 0. Therefore, considering |λ| → ∞ we get µ(A) = 0, that is, µ ≪ ν and µλa ≪ ν. 3. If a non-negative measure µ is pseudo-differentiable in a direction a ∈ Jµr and r ˜ (Da µ) ≪ µ, then lµr (a, x) := (D˜ ra µ)(dx)/µ(dx) is called the logarithmic pseudo-derivative of quasi-invariant measure µ in a direction a of order r, where r > 0. Define the sequence of logarithmic pseudo-derivatives lnr of µn , where µn denotes the projection of the measure µ on finite dimensional over K subspaces H(n), Pn : X → H(n) is the projection, H(n) ⊂ H(n+1) S for each n ∈ N, n H(n) is everywhere dense in X. Theorem. The sequence of functions lnr (Pn a, Pn x) is the martingale. The logarithmic pseudo-derivative lµr (a, x) in a direction a ∈ X of order r > 0 exists if and only if the sequence lnr (Pn a, Pn x) is uniformly integrable. Moreover, lµr (a, x) = lim lnr (Pn a, Pn x) (mod µ) and for r = 1 there is the equality: n→∞

Z

X

[ f (x + λa) − f (x)] µ(dx) =

Z h X

i ξ=λ D˜ −1 f (x + ξa) |ξ=0 lµ (a, x)µ(dx). ξ

(1) (2)

Proof. Let µ be pseudo-differentiable along a1 and a2 . Evidently, g(x, y, r) = g(y, x, r) = g(x − y, 0, r) for all x 6= y (see § 4.1). In view of 4.4(i, ii) we have Z

=

Z Z K

=

X

K

=

 f (x) [µ(−λ(a1 + a2 ) + dx) − µ(dx)] g(λ, 0, r)v(dλ)

Z Z

Z Z K

+

X

X

f (x)D˜ ra1 +a2 µ(dx)

X

 [ f (x + λ(a1 + a2 )) − f (x)]µ(dx) g(λ, 0, r)v(dλ)

 [ f (x + λ(a1 + a2 )) − f (x + λa1 )]µ(dx) g(λ, 0, r)v(dλ)

Z Z K

X

 [ f (x + λa1 ) − f (x)]µ(dx) g(λ, 0, r)v(dλ).

(3)

64

Sergey V. Ludkovsky

The Banach space X is totally disconnected. The integrals on the right side exist for each continuous simple function f , since χB (x + y) = χB (x) for each clopen ball B(X, x0 , R) := {x ∈ X : |x − x0 | ≤ R} of radius R > 0 with |y| < R, where χA denotes the characteristic function of a set A, χA (x) = 1 for each x ∈ A, χA (x) = 0 for each x ∈ / A. The space of continuous simple functions is dense in Cb0 (X, C) ∩ L1 (µ), consequently, (3) has the continuous extension to the C-linear functional on Cb0 (X, C). Thus D˜ ra1 +a2 µ exists. Suppose that D˜ ra µ exists and ξ ∈ K, ξ 6= 0, then  Z Z Z r ˜ f (x) [µ(−λξa + dx) − µ(dx)] g(λ, 0, r)v(dλ) (4) f (x)Dξa µ(dx) = X

X

K

=

Z Z K

X

 [ f (x + λξa) − f (x)]µ(dx) g(λ, 0, r)v(dλ) Z Z

= [1/modK (ξ)]

K

X

 [ f (x + za) − f (x)]µ(dx) g(λ, 0, r)v(dz)

= [1/modK (ξ)]

Z

X

f (x)D˜ ra µ(dx),

hence modK (ξ)D˜ rξa µ = D˜ ra µ exists for each 0 6= ξ ∈ K, where D˜ r0 µ = 0. Thus we have demonstrated, that the set Xµr of all vectors a in X for which D˜ ra µ exists is the K-linear subspace in X. If lµr (a, x) exists, then Z

X

f (x)D˜ ra µ(dx)

=

Z Z K

=

X

Z

 [ f (x + λa) − f (x)]µ(dx) g(λ, 0, r)v(dλ)

X

f (x)lµr (a, x)µ(dx).

(5)

This implies that the set Lµr of all those a ∈ X for which the logarithmic pseudo-derivatives of order r exists is the K-linear subspace in X. Particularly, if f (x) = φ(PL x) is a cylindrical function, where PL : X → L is the projection operator, for example, for L = H(n), a ∈ Lµr and z = PL a, then  Z Z [φ(x + zu) − φ(x)]µL (dx) g(z, 0, r)v(dz) L

K

=

Z Z K

=

X

Z

X

 [ f (x + zu) − f (x)]µ(dx) g(z, 0, r)v(dz) f (x)D˜ ra µ(dx) =

Z

L

φ(x)D˜ rz µL (dx),

(6)

hence D˜ rz µL (dx) ≪ µL and inevitably µL has the logarithmic pseudo-derivative along z of order r > 0. In view of Theorem 3.2.4 we get: d(D˜ rz µL (x))/dµL (x) =

Z

X

lµr (a, z)µ(dz, B L |PL−1 x),

(7)

Real-Valued Measures

65

where µ(dz, B L |PL−1 x) is the conditional measure corresponding to µ relative to the σalgebra B L of all sets PL−1 A, where A ∈ B f (L) is a Borel set in L. Consider the sequence of logarithmic pseudo-derivatives lnr of µn , where µn denotes the projection of the measure µ on H(n), n ∈ N. In view of Formula (7) d(D˜ ra(n) µH(n) (Pn x))/dµH(n) (Pn x) =

Z

H(n)

H(m)

lµr (a(m), z)µH(m) (dz, B H(m) |x)

(8)

H(n)

for each n < m, where µH(m) (dz, B H(m) |x) is the conditional measure corresponding to H(n)

µH(m) relative to the σ-algebra B H(m) of all sets H(m) ∩ Pn−1 A, where A ∈ B f (H(m)) is a Borel set in H(m), Pn : X → H(n) is the projection operator, a(n) = Pn a. Moreover, H(n) µH(m) (A, B H(m) |x) = µ(Pm−1 A, B H(m) |Pm−1 x). Therefore, (8) takes the form lnr (a(n), x) =

Z

H(m)

lmr (a(m), z)µ(dz, B H(n) |x).

(9)

The function lnr is B H(n) -measurable, hence due to §§ 2.37 and 2.39 Z

X

lnr (a(n), Pn x)ψ(x)µ(dx) = =

Z

Z Z X

H(n)

H(m)

lmr (a(m), z)µH(m) (dz, B H(m) |x)ψ(x)µ(dx)

lmr (a(m), Pm x))ψ(x)µ(dx)

H(m)

for each B H(n) -measurable function ψ(x), where x(m) = Pm x. This means that lnr is the martingale. In accordance with Formula (7) we have lnr (a(n), Pn x)

Z

=

X

lµr (a, z)µ(dz, B H(n) |x).

(10)

Consider the function gN (t) so that gN (t) = 0 for t < N, gN (t) = t − N for reach t ≥ N. To prove the uniform integrability it is sufficient to show that lim sup

N→∞ n

Z

X

gN (|ln (a(n), Pn x)|)µ(dx) = 0.

(11)

The function gN (t) is downward convex so by the Jensen inequality gN (|lnr (a(n), Pn x)|) ≤ Z

X

Z

X

gN (|lµr (a, z)|)µ(dz, B H(n) |x), hence

gN (|lnr (a(n), Pn x)|)µ(dx) ≤ =

Z

X

Z Z X

X

gN (|lµr (a, z)|)µ(dz, B H(n) |x)µ(dx)

gN (|lµr (a, x)|)µ(dx).

(12)

Since gN (t) ≤ |t| for each t ∈R R, X |lµ (a, x)|µ(dx) < ∞ and limN→∞ gN (|lµr (a, x)|) = 0 for µ-almost all x, then limN→∞ X gN (|lµr (a, x)|)µ(dx) = 0. Combining this with (12) we get (11). In view of Theorem 2.42 the limit limn→∞ lnr (Pn a, Pn x) = lµr (a, x) exists for µ-almost all x. R

66

Sergey V. Ludkovsky

Take an arbitrary continuous on H(n) function φ and put f (x) = φ(Pn x), then due to (5 − 7) we infer: Z

[ f (x + a) − f (x)]µ(dx) =

Z Z X

X

K

f (x + za)lnr (Pn a, Pn x)v(dz)µ(dx).

(13)

The uniform integrability of ln permits to take the limit by n tending to the infinity, that gives the equality: Z

[ f (x + a) − f (x)]µ(dx) = X

Z Z X

K

f (x + za)lµr (a, x)v(dz)µ(dx).

(14)

This functional is C-linear and continuous by f ∈ Cb0 (X, C), so (14) is satisfied for each f ∈ Cb0 (X, C) and inevitably lµr (a, x) = d((D˜ ra µ)/dµ)(x). Applying §§ 4.2-4.4 we get Formula (2). 4.7. Theorem. Let X be a Banach space over K, | ∗ | = modK (∗) with a probability quasi-invariant measure µ : B f (X) → R and it is satisfied Condition 3.21(i), suppose µ is pseudo-differentiable and (viii) Jb µ ⊂ T ”Jµ , (Ut : t ∈ B(K, 0, 1)) is a one-parameter family of operators such that Conditions 3.25(i − vii) are satisfied with the substitution of Jµ onto Jµb uniformly by t ∈ B(K, 0, 1), Jµ ⊃ T ′ X, where T ′ , T ” : X → X are compact operators, ker(T ′ ) = ker(T ”) = 0. Moreover, suppose that there are sequences (ix) [k(i, j)] and [k′ (i, j)] with i, j ∈ N, limi+ j→∞ k(i, j) = limi+ j→∞ k′ (i, j) = −∞ and n ∈ ′ N such that |T ”i, j −δi, j | < |Ti,′ j −δi, j |pk(i, j) , |Ui, j −δi, j | < |T ”i, j −δi, j |pk (i, j) and |(U −1 )i, j − ′ δi, j | < |T ”i, j − δi, j |pk (i, j) for each i + j > n, where Ui, j = e˜iU(e j ), (e j : j) is orthonormal basis in X. Then for each f ∈ Cb (X) there is defined  Z Z −1 f (x)[µ(Ut (dx)) − µ(dx)] g(t, 0, b)v(dt) (i) l( f ) = B(K,0,1)

X

and there exists a measure ν : B f (X) → C with a bounded total variation [particularly, for b ∈ R it is such that ν : B f (X) → R] and (ii) l( f ) =

Z

f (x)ν(dx), X

where ν is independent from f and may be dependent on (Ut : t), ν =: D˜ Ub ∗ µ. Proof. From the proof of Theorem 3.25 it follows that there exists a se(q) quence Ut of polygonal operators converging uniformly by t ∈ B(K, 0, 1) to Ut and equicontinuously by indices of matrix elements in L1 . Then there exists R R limq→∞ lim j→∞ B(K,0,1)\B(K,0,p− j [ X f (Ut−1 (x)) − f (x)]g(t, 0, b)µ(dx)]v(dt) for each f ∈ Cb (X).R From conditions (viii, ix), the Fubini and Lebesgue theorems it follows that for νλ := B(K,0,1)\B(K,0,|λ|) [µ(Ut−1 (A)) − µ(A)]g(t, 0, b)v(dt) for A ∈ B f (X) for each c > 0 there exists a compact Vc ⊂ X and δ > 0 such that |νλ |(X \ Vc ) < c. Indeed, Vc and δ > 0 may be chosen due to pseudo-differentiability of µ, §§ 2.35, 3.9, 3.23, Formula (i), 3.21(i) and due to continuity and boundednessy (on B(K, 0, 1) ∋ t) of |det Ut′ (Ut−1 )(x))|K satisfying the following conditions Ut−1 (Vc ) ⊂ Vc and k(X \ Vc ) △ (Ut−1 (X \ Vc ))kµ = 0 for each |t| < δ, since Vc = Y ( j) ∩Vc are compact for every j. At the same time closed subsets Y ( j) ⊂ X for

Real-Valued Measures

67

(q)

Ut may be chosen independent from t and in § 2.35 operator Sc may be chosen symmetric, T X ⊃ Jµ ⊃ T ′ X, where T is a compact operator. Evidently, conditions of type (ix) are carried out for V ( j, x) and U( j, x) uniformly by j. Repeating proofs 3.25 and 4.4 with the use of Lemma 2.5 for the family (Ut : t) we get formulas (i, ii).

1.5.

Convergence of Measures

Different types of convergence of measures were considered in [BS90, Con84, SF76, Top74, Top76]. The definitions and theorems given below are taking into account the properties of quasi-invariance and pseudo-differentiability of measures. 5.1. Definitions, notes and notations. Let S be a normal topological group with the small inductive dimension ind(S) = 0, S′ be a dense subgroup, suppose their topologies are τ and τ′ correspondingly, τ′ ⊃ τ|S′ . Let G be an additive Hausdorff left-R-module, where R is a topological ring, R ⊃ B f (S) be a σ-ring for real-valued measures, M(R,G) be a family of measures with values in G, L(R,G, R) be a family of quasi-invariant mea−1 sure µ : R → G with ρµ (g, x) × µ(dx) := µg (dx) =: µ(gdx), R × G → G be a continuous left action of R on G such that ρµ (gh, x) = ρµ (g, hx)ρµ (h, x) for each g, h ∈ S′ and x ∈ S. Particularly, 1 = ρµ (g, g−1 x)ρµ (g−1 , x), that is, ρµ (g, x) ∈ Ro , where Ro is a multiplicative subgroup of R. Moreover, zy ∈ L for z ∈ R0 with ρzµ (g, x) = zρµ (g, x)z−1 and z 6= 0. We suppose that topological characters and weights S and S′ are countable and each open W in S′ is precompact in S. Let P” be a family of pseudo-metrics in G generating the initial uniformity such that for each c > 0 and d ∈ P” and {Un ∈ R : n ∈ N} with ∩{Un : n ∈ N} = {x} there is m ∈ N such that d(µg (Un ), ρµ (g, x)µ(Un )) < cd(µ(Un ), 0) for each n > m, in addition, a limit ρ is independent µ-a.e. on the choice of {Un : n} for each x ∈ S and g ∈ S′ . Consider a subring R′ ⊂ R, R′ ⊃ B f (S) such that ∪{An : n = 1, . . . , N} ∈ R′ for An ∈ R′ with N ∈ N and S′ R′ = R′ . Then L(R, G, R; R′ ) := {(µ, ρµ (∗, ∗)) ∈ L(R, G, R) : µ − R′ − is regular and for each s ∈ S there are An ∈ R′ , n ∈ N with s = ∩(An : n), {s} ∈ R′ }. For pseudo-differentiable measures µ let S” ⊂ S′ , S” be a dense subgroup in S, τ′ |S” is not stronger than τ” on S” and there exists a neighborhood τ” ∋ W ” ∋ e in which are dense elements lying on one-parameter subgroups (Ut : t ∈ B(K, 0, 1)). We suppose that µ is induced from the Banach space X over K due to a local homeomorphism of neighborhoods of e in S and 0 in X as for the case of groups of diffeomorphisms [Lud96] such that is accomplished Theorem 4.7 for each U∗ ⊂ S” inducing the corresponding transformations on X. In the following case S = X we consider S′ = Jµ and S” = Jµb with Re(b) > 0 such (b)

(b)

that Mµ ⊃ Jµ ⊂ (Tµ X)∼ , Jµb ⊂ (Tµ X)∼ with compact operators Tµ and Tµ , ker(Tµ ) = (b)

ker(Tµ ) = 0 and norms induced by the Minkowski functional PE for E = Tµ B(X, 0, 1) and (b) E = Tµ B(X, 0, 1) respectively. We suppose further that for pseudo-differentiable measures G is equal to C or R. We denote P(R, G, R,U∗ ; R′ ) := [(µ, ρµ , ηµ ) : (µ, ρµ ) ∈ L(R, G, R; R′ ), µ is pseudo-differentiable and ηµ (t,U∗ , A) ∈ L1 (K, v, C)], where ηµ (t,U∗ , A) = j(t)g(t, 0, b)[µh (Ut−1 (A) − µh (A)], j(t) = 1 for each t ∈ K for S = X; j(t) = 1 for t ∈ B(K, 0, 1), j(t) = 0 for |t|K > 1 for a topological group S that is not a Banach space X over K, v is the Haar measure on K

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with values in [0, ∞), (Ut : t ∈ B(K, 0, 1)) is an arbitrary one-parameter subgroup. On these spaces L (or P) the additional conditions are imposed: (a) for each neighborhood (implying that it is open) U ∋ 0 ∈ G there exists a neighborhood S ⊃ V ∋ e and a compact subset VU , e ∈ VU ⊂ V , with µ(B) ∈ U (or in addition D˜ Ub ∗ µ(B) ∈ U) for each B, R ∋ B ∈ B f (S \VU ); (b) for a given U and a neighborhood R ⊃ D ∋ 0 there exists a neighborhood W , S′ ⊃ W ∋ e, (pseudo)metric d ∈ P” and c > 0 such that ρµ (g, x) − ρµ (h, x′ ) ∈ D (or D˜ Ub ∗ (µg − µh )(A) ∈ U for A ∈ B f (VU ) in addition for P) whilst g, h ∈ W , x, x′ ∈ VU , d(x, x′ ) < c, where (a,b) is satisfied for all (µ, ρµ ) ∈ L (or (µ, ρµ , ηµ ) ∈ P) equicontinuously in (a) on V ∋ Ut ,Ut−1 and in (b) on W and on each VU for ρµ (g, x) − ρµ (h, x′ ) and D˜ Ub ∗ (µg − µh )(A). These conditions are justified, since for a Gaussian measure ν on a Hilbert space Z over C or R (or measures given above for a non-Archimedean Banach space Z over K) there exists a subspace Z ′ dense in Z such that ν is quasi-invariant relative to Z ′ . Moreover, in view of Theorem 26.2[Sko74] (or Theorems 3.20, 3.25, 4.3 and 4.7 in the non-Archimedean case respectively) there exists a subspace Z” dense in Z ′ such that for each ε > 0 and each ∞ > R > 0 there are r > 0 and δ > 0 with |ρν (g, x) − ρν (h, y)| < ε for each kg − hkZ” + kx − ykZ < δ, g, h ∈ B(Z”, 0, r), x, y ∈ B(Z, 0, R), where Z” is the Banach space over C or R or K respectively. For a group of diffeomorphisms of a Hilbert manifold over R or over a non-Archimedean Banach manifold we have an analogous continuity of ρµ for a subgroup G” of the entire group G (see [Lud96, Lud99t, Lud00a, Lud02b]). By Mo we denote a subspace in M, satisfying (a). Henceforth, we imply that R′ contains all closed subsets from S belonging to R, where G and R are complete. For µ : B f (S) → G by L(S, µ, G) we denote the completion of a space of continuous R f : S → G such that k f kd := suph∈Cb (S,G) d( S f (x)h(x) µ(dx), 0) < ∞ for each d ∈ P”, where Cb (S, G) is a space of continuous bounded functions h : S → G. We suppose that for each sequence ( fn : n) ⊂ L(S, µ, G) for which g ∈ L(S, µ, G) exists with d( fn (x), 0) ≤ d(g(x), 0) for every d ∈ P”, x and n, that fn converges uniformly on each compact subset V ⊂ S with |µ|(V ) > 0. In the cases G = C it coincides with L1 (S, µ, C) correspondingly, hence this supposition is the Lebesgue theorem. By Y (v) we denote L1 (K, v, C). Now we may define topologies and uniformities with the help of corresponding bases ′ ′ ′ (see below) on L ⊂ GR × RSo ×S =: Y (or P ⊂ GR × RSo ×S × GS ×K×R =: Y ), Ro ⊂ R \ {0}. There are the natural projections π : L (orP) → Mo , π(µ, ρµ (∗, ∗) (∨ ,ηµ )) = µ, ξ : L (∨ ′ ′ P) → RS ×S , ξ(µ, ρµ , (∨ ηµ )) = ρµ , ζ : P → GS ×K×R , ζ(µ, ρµ , ηµ ) = ηµ . Let H be a filter on L or P, U = U ′ ×U” or U = U ′ ×U” ×U”′ , U ′ and U” be elements of uniformities on G, R and Y (v) correspondingly, τ′ ∋ W ∋ e, τ ∋ V ⊃ VU ′ ∋ e, VU ′ is compact. By [µ] we denote (µ, ρµ ) for L or (µ, ρµ , ηµ ) for P, Ω := L ∨ P, [µ](A,W,V ) := [µg (A), ρµ (g, x), ∨ ηµg (t,U∗ , A)| g ∈ W, x ∈ V, ∨ t ∈ K]. We consider A ⊂ R, then W(A,W,VU ′ ;U) := {([µ], [ν]) ∈ Ω2 |([µ], [ν])(A,W,VU ′ ) ⊂ U};

(1)

W(S;U) := {([µ], [ν]) ∈ Ω2 |{(B, g, x) : ([µ], [ν])(B, g, x)) ∈ U} ∈ S},

(2)

where S is a filter on R × S′ × Sc , Sc is a family of compact subsets V ′ ∋ e. W(F,W,V ;U) := {([µ], [ν]) ∈ Ω2 |{B : ([µ], [ν])(B, g, x) ∈U, g ∈ W, x ∈V } ∈ F},

(3)

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where F is a filter on R (compare with § 2.1 and 4.1[Con84]); W(A, G;U) := {([µ], [ν]) ∈ Ω2 | {(g, x) : ([µ], [ν])(B, g, x) ∈ U, B ∈ A} ∈ G},

(4)

where G is a filter on S′ × Sc ; suppose U ⊂ R × τ′e × Sc , Φ is a family of filters on R × S′ × Sc or R × S′ × Sc ×Y (v) (generated by products of filters ΦR × ΦS′ × ΦSc on the corresponding spaces), U′ be a uniformity on (G, R) or (G, R,Y (v)), F ⊂ Y . A family of finite intersections of sets W(A,U) ∩ (F × F) (see (1)), where (A,U) ∈ U × U′ (or W(F,U) ∩ (F × F) (see (2)), where (F,U) ∈ (Φ × U′ ) generate by the definition a base of uniformity of U-convergence (Φ-convergence respectively) on F and generate the corresponding topologies. For these uniformities are used notations (i) FU and FΦ ;

FR×W ×V is for F with the uniformity of uniform convergence

on R ×W ×V , where W ∈ τ′e , V ∈ Sc , analogously for the entire space Y ; (ii) FA denotes the uniformity (or topology) of pointwise convergence for A ⊂ R × τ′e × Sc =: Z, for A = Z we omit the index (see formula (1)). Henceforward, we ˜ use H′ instead of H in 4.1.24[Con84], that is, H′ (A, R)-filter on R generated by the base ′ ′ ′ ˜ ˜ ˜ is closed relative to the finite [(L ∈ R : L ⊂ A \ K ) : K ∈ R, K ⊂ A], where R ⊂ R and R unions. For example, let S be a locally K-convex space, S′ be a dense subspace, G be a locally L-convex space, where K, L are fields, R = B(G) be a space of bounded linear operators on G, Ro = GL(G) be a multiplicative group of invertible linear operators. Then others possibilities are: S = X be a Banach space over K, S′ = Jµ , S” = Jµb as above; S = G(t), S′ ⊃ S” are dense subgroups, G = R be the field R, M be an analytic Banach manifold over K ⊃ Qp (see [Lud96]). The rest of the necessary standard definitions are recalled further when they are used. 5.2. Lemma. Let R be a quasi-δ-ring with the weakest uniformity in which ˆ C (R, S′ × Sc ). Then L(R, G, R, R′ )Φ (or each µ ∈ M is uniformly continuous and Φ ⊂ Φ ′ P(R, G, R,U∗ ; R )Φ ) is a topological space on which Ro acts continuously from the right. ˆ := Φ(X) ˆ Proof. We recall that Φ denotes a family of filters F on X such that for each mapping f : Φ → B(X) with f (Σ) ⊂ Σ for each Σ ∈ Φ there exists a finite subset Ψ ⊂ Φ S such that Σ∈Ψ f (Σ) ∈ F, where B(X) denotes a family of all subsets in X, ΦC (X) =: ΦC be a family of Cauchy filters on X. In view of proposition 4.2.2[Con84] the space of measures M(R, G; R′ )Φ is a topological left-R-module. On the other hand, ρλµ (a, x) = λρµ (a, x)λ−1 for λ ∈ Ro , from the continuity of the inversion λ 7→ λ−1 in Ro and the multiplication in R for each entourage of the diagonal U” in R there exists an entourage of a diagonal U”1 in R and an open neighborhood U4 ∋ λ in Ro such that βU”1 β−1 ⊂ U” for each β ∈ U4 . Choosing Ψ ⊂ T ˜ and Φ for a given F we find U”1 (Σ) and U4 (Σ) for each Σ ∈ Ψ, then Σ∈Ψ U”1 (Σ) =: U” T U˜ 4 = Σ∈Ψ U4 (Σ) are entourages of the diagonal and generate a neighborhood of λ. This shows the continuity by λ for L and P, since for P is satisfied: PD(b, λ f (x)) = λPD(b, f (x)) due to definition 4.1 for pseudo-differentiable f . ˆ 4 -filter on Mo (R,G; R′ ), {An } be a disjoint Θ(R)5.3. Proposition. (1). Let T be a Φ sequence, Σ be the elementary filter on R generated by {An : n ∈ N} and φ : Mo × R → G with φ(µ, A) = µ(A). Then φ(T × Σ) converges to 0.

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(2). Moreover, let U be a base of neighborhoods of e ∈ S′ , φ : L → G × R, φ(µ, A, g) := where x ∈ A. Then (0, 1) ∈ lim φ(T × Σ × U). (3). If T is a Φ4 -filter on P(R, G, R,U∗ ; R′ ), ψ(µ, B, g,t,U∗ ) = [µ(B); ρµ (g, x); ηµg (t,U∗ , B)], then (0, 1, 0) ∈ lim ψ(T × Σ × U) for each given U∗ ∈ S”, where Σ and U as in (1, 2). Proof. We recall that for a set X, Θ-net in X is a net (I, f ) in X such that for each increasing sequence (in : n ∈ N) ⊂ I is satisfied ( f (in ) : n) ∈ Θ. A sequence (xn : n ∈ N) ⊂ X is called a Θ-sequence, if f : N → X, f (n) = xn , (N, f ) is a Θ-net. Then Φ(Θ, X) =: Φ(Θ) denotes a family of filters on X of the form f (F), where (I, f ) is a Θ-net in X, F is a filter ˆ of sections [(λ ∈ I : λ ≥ i) : i ∈ I] for a directed set I, Φ(Θ) := Φ(Θ)ˆ. For a topological space X, Θ j (X) denotes a family of sequences having converging subsequences for j = 1 or having limit points whilst j = 2. For a uniform space X, Θ3 (X) denotes a family of sequences having Cauchy subsequences Θ4 := Θ2 ∪ Θ3 , Φ j (X) := Φ(Θ j (X)). ThenR(1) follows from 4.2.6[Con84]. (2) From the restrictions (a,b) in 5.1 it follows that µg (A) = A ρµ (g, x)µ(dx). If in addition lim µ(F(A; R′ )) = µ(A), then lim µ(F(X \ A, R′ )) = µ(X \ A), since by the definition 4.1.24[Con84] for each τG ∋ D ∋ 0 for A ∈ R there exists τS ∋ V ⊃ A with µ(V ) − µ(A) ∈ D, where F(A, R′ ) is a filter generated by A and R′ . Due to the Radonian property of µ and µg we get limT×Σ µg (A) = 0. (3). Additionally to (1,2) it remains to verify that ηµ converges. In view of Theorems 4.4 and 4.7 (or §5.1) a pseudo-differential of order b : D˜ Ub ∗ µ = ν is a measure for pseudoˆ 4 -condition we get (0, 1, 0) ∈ lim ψ(T × Σ × U) for a differentiable µ. Due to (1) and the Φ given U∗ , since in §§ 4.1 and 5.1 the integral is by t ∈ K for the Banach space X over K and by B(K, 0, 1) correspondingly for S that is not the Banach space. ˆ 4 -filter on L (or P) with the topology F (see § 5.1.(ii)), 5.4. Proposition. Let H be a Φ ′ ′ A ∈ R, τG ∋ U ∋ 0, H (A, R ) ∈ Ψ f (R). Then there are L ∈ H, K˜ ∈ R′ and an element ˜ ⊂ A, L = [(µ, ρµ (g, x)) : M := πMo (L) ∋ of the uniformity U for LR′ or PR′ such that K µ, πτ′e (L) =: W ∋ g (or (µ, ρµ , ηµ (∗, ∗,U∗ )) and additionally D˜ Ub ∗ µ = PD(b, ηµ ))], e ∈ W ∈ τ′ , µg (B)−νh (C) ∈ U (or in addition (D˜ Ub ∗ µg (B))−(D˜ Ub ∗ νh (C)) ∈ U) for K˜ ⊂ B ⊂ A, K˜ ⊂ C ⊂ A for each ([µ], [ν]) ∈ L¯ 2 ∩ U, where L¯ := cl(L, LR′ ) (or cl(L, PR′ )), πMo is a projector from L into Mo . ˆ Proof. We recall that Ψ(R) := Φ(Θ(R)), where Θ(R) is a family of sequences (An : n ∈ N) ⊂ R for which there exists Ω ∈ Σ(R) with (n ∈ N : An ∈ Ω) is infinite, Σ(R) := S [( n∈J An : J ∈ B(N)) : (An : n ∈ N) ∈ Γ(R)], Γ(R) is a family of disjoint sequences (An : S n ∈ N) ⊂ R for which [ n∈J An : J ⊂ N] ⊂ R]. A ring of sets Z is called a quasi-δ-ring, if S each disjoint sequence (An : n ∈ N) from R the union of which n An = A is contained in a set B ∈ R, A ⊂ B, has a subsequence (An j : j ∈ N) ∈ Γ(R). From Proposition 4.2.7[Con84] and the Radonian property of measures we have µ(B) − ¯ M) ∩ π (U). Then for each element D′ of the uniformity ν(C) ∈ U ′ for each (µ, ν) ∈ (M × M0 on R there are d ∈ P”, c > 0 and L such that ρµ (g, x) − ρν (h, x′ ) ∈ D′ for g, g′ ∈ W , µ, ν ∈ M, ¯ := cl(M, E(R, G; R′ ) ′ ) is the closure of M ⊂ E(R, G; R′ ) ˆ 4 , where M d(x, x′ ) < c, since H ∈ Φ R ′ ′ in E(R, G; R )R′ , E(R, G; R ) := [µ ∈ E(R, G) : µ is R′ -regular ], that is, µ(F(A, R′ )) converges to µ(A) for each A ∈ R (simply regular, if R′ consists of closed subsets and conditions in definition 11.34[HR79] are satisfied). Then F(A, R′ ) is a filter on R generated by a base [(B ∈ R : K˜ ⊂ B ⊂ A) : K˜ ⊂ A, K˜ ∈ R′ ], E(R, G) is a set of exhaustive additive maps µ, that (µg (A), ρµ (g, x)),

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is, µ(An ) converges to 0 for each (An : n ∈ N) ∈ Γ(R). Then µg (B)−µ(B) ∈ U ′ , νh (C)−ν(C) ∈ U ′ and for 3U ′ ⊂ U we get 5.4 for L. From Theorems 4.4 and 4.7 (or § 5.1), the Egorov conditions and the Lebesgue theorem we get 5.4 for P, since µ are probability measures and LR′ (or PR′ ) correspond to uniformity from § 5.1.(ii) with A = R′ × τ′e × Sc . Indeed, µg (A) − νh (A) = (µg (A) − µg (VU ′ )) + (µg (VU ′ ) − νh (VU ′ )) + R (νh (VU ′ ) − νh (A)), µg (A) = A ρµ (g, x)µ(dx) for each A ∈ B f (S), for each τG ∋ U ′ ∋ 0 there exists a compact subset VU′ ⊂ A with µg (B) ∈ U ′ for each B ∈ B f (A \VU ′ ) and the same for νh (due to the Condition in 5.1 that R′ contains B f (S)). From the separability of S, S′ and the equality of their topological weights to ℵ0 , restrictions 5.1(a,b) it follows that there exists a sequence of partitions Zn = [(xm , Am ) : m, xm ∈ Am ] for each A ∈ B f (S), Ai ∩ A j = 0/ for S each i 6= j, m Am = A, Am ∈ B f (S), such that limn→∞ (µg (A) − ∑ j ρµ (g, x j )µ(A j )) = 0 and the same for ν, moreover, for VU ′ each Zn may be chosen finite. Then there exists W ∈ τ′e with W × (S \ V 2 ) ⊂ (S \ V ), τe ∋ V ⊂ V 2 , νg (B) and µg (B) ∈ U ′ for each B ∈ B f (S \ V 2 ) (for G = R) and g ∈ W (see 5.1.(a)). Then from A = [A ∩ (S \ V 2 )] ∪ [A ∩ V 2 ] and the existence of compact VU′ ′ ⊂ V with µ(E) ∈ U ′ for each E ∈ B f (V \ VU′ ′ ) and the same for ν, moreover, (VU′ ′ )2 is also compact, it follows that µg (B) − νh (C) ∈ U for 9U ′ ⊂ U, since R′ ⊃ B f (S), where W satisfies the following condition µg (VU′ ′ ) − νh (VU′ ′ ) ∈ U ′ for VU′ ′ ⊂ V 2 due to 5.1.(b), µ(B) − ν(C) ∈ U ′ , WVU′ ′ ⊂ (VU′ ′ )2 due to precompactness of W in S. Since pseudo-differentiable measures are also quasi-invariant, hence for them Proposition 5.4 is true. Now let [µ] ∈ lim H, A ∈ B f (S), then ηµ ∈ lim ζ(H) in Y (v) and there exists a seR quence ηµn such that K ηµn (λ,U∗ , A)v(dλ) = D˜ Ub ∗ µn (A) due to §§ 4.4, 4.7 or 5.1 and R limn→∞ D˜ Ub ∗ µn (A) = K ηµ (λ,U∗ , A)v(dλ) =: κ(A) due to the Lebesgue theorem. From / B ∈ B f (S) and the Nikodym ηµ (λ,U∗ , A ∪ B) = η(λ,U∗ , A) + η(λ,U∗ , B) for A ∩ B = 0, theorem [Con84] it follows that ν(A) is the measure on B f (S), moreover, κ(A) = D˜ Ub ∗ µ(A). R Since µg (A) = A ρµ (g, x)µ(dx) for A ∈ B f (S) for g ∈ S′ , then ηµg (λ,U∗ , A) = j(λ)g(λ, 0, b)[µg (A) − µg (Uλ−1 A)] = j(λ)g(λ, 0, b)

Z

A

ρµ (g, x)[µ(dx) − µUλ (dx)]

and in view of the Fubini theorem there exists  Z Z U b g λ ρµ (g, x) j(λ)g(λ, 0, b)[µ(dx) − µ (dx) v(dλ), D˜ U∗ µ (A) = A

K

where j(t) = 1 for S = X and j(t) is the characteristic function of B(K, 0, 1) for S that is not the Banach space X. Then µ-a.e. D˜ Ub ∗ µg (dx)/D˜ Ub ∗ µ(dx) coincides with ρµ (g, x) due to ˆ 4 -filter in P. Then 5.1.(a,b), hence, (D˜ Ub ∗ µg , ρµg ) generate the Φ4 -filter in L arising from the Φ we estimate D˜ Ub ∗ (µg − νh )(A) as above µg (A) − νh (A). Therefore, we find for the Φ4 -filter corresponding L, since there exists δ > 0 such that Uλ ∈ W for each |λ| < δ. For Φ4 -filter we use the corresponding finite intersections W1 ∩ · · · ∩ Wn = W , where W j correspond to the Φ4 -filters H j . ˆ 4 -filter in L, U is an element 5.5. Corollary. If {H′ (A, R′ ) : A ∈ R} ⊂ Ψ f (R), T is a Φ of uniformity in L (or P), then there are L ∈ T and an element V of the uniformity in LR′ (or PR′ ) such that L¯ 2 ∩ V ⊂ L¯ 2 ∩ U.

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Proof. In view of Corollary 4.2.8[Con84] and Theorem 5.4 above for UM = πM × πM U there exists an entourage of the diagonal (see about uniform spaces in [Eng86]), VM in E(R, G; R′ )R′ , such that M¯ 2 ∩VM ⊂ M¯ 2 ∩ UM , where πM : L → M (or P → M) is a projector, M(R, G) is a set of measures on R with values in G, M(R, G; R′ ) := [µ ∈ M(R, G) : µ is R′ regular ]. Since M(R, G; R′ )R′ ⊂ E(R, G; R′ )R′ , then there exists an entourage of the diagonal, V in LR′ (or PR′ ), such that πM × πMV ⊂ VM ∩ M¯ 2 . 5.6. Lemma. Let R be a quasi-σ-ring directed by the inclusion, U be an upper diˆ 4 -filter in L (or P), 0 ∈ U ∋ τG . Then there are M ∈ H, A ∈ U, rected subset in R, H be a Φ an element of the uniformity U ′ in LR′ (or PR′ ) such that µg (B) − νh (C) ∈ U (or in addition D˜ Ub ∗ µg (B) − D˜ Ub ∗ νh (C) ∈ U) for each ([µ], [ν]) ∈ L¯ 2 ∩ U ′ , g, h ∈ W , W is defined by a projection of U ′ onto S′ , B,C ∈ U with A ≤ B, A ≤ C. Proof. For each C and E ∈ U by the definition there are F ∈ U with C ≤ F and E ≤ F. There are open subsets P and V in S and W ∈ τ′e with W P ⊂ P2 ⊂ V such that (µ−ν)(B) ∈ U ¯2 for each B ⊂ S \ P and (µ, ν) ∈ ¯M ∩ πMo (U ′ ). Indeed, πMo H is a base of a filter T in Mo . Therefore, g(B ∩ S \ P2 ) ⊂ S \ P for g ∈ W , consequently, [µg − νg ](B ∩ S \ P2 ) ∈ U. In view of § 5.1.(a,b) for D,V there are W , c > 0, d ∈ P” such that ρµ (g, x) − ρν (h, x′ ) ∈ D for g, h ∈ W and d(x, x′ ) < c, x, x′ ∈ Vc , consequently, µg (B) − νh (C) = [µg (B ∩ P2 ) − νh (C ∩ P2 )] + [µg (B \ P2 ) − νh (C \ P2 )] ∈ 3(DU +U). Modifying the proof of § 4.2.9 [Con84] we get the statement of this lemma for L. For P an estimate of D˜ Ub ∗ µg (B) − D˜ Ub ∗ νh (C) may be done analogously to the proof of Proposition 5.4. ˆ 4 -filter in L (or P), {An : n ∈ N} ∈ Γ(R), τG ∋ U ∋ 0. Then 5.7. Theorem. Let H be a Φ there are L ∈ H, M ∈ B(N) and an element U of the uniformity in L such that µg (∪(An : n ∈ S S M ′ ) − νh (∪(An : n ∈ M”) ∈ U (or in addition (D˜ Ub ∗ µg ( n∈M′ ) − D˜ Ub ∗ νh ( n∈M” An )) ∈ U for each ([µ], [ν]) ∈ L¯ 2 ∩ U and M ′ , M” ∈ B(N) with M ∩ M ′ = M ∩ M”. If {H′ (A, R′ )} ⊂ Ψ f (R), then U may be chosen as an element of the uniformity in LR′ (or PR′ ), where L¯ := cl(L, L) (or L¯ := cl(L, P)). Proof. We recall that B f (N) denotes the family of finite subsets in N, Ψ f (R) := ˆ f (R)), Θ f (R) is a family of sequences (An : n ∈ N) ⊂ R for which there exists Φ(Θ S Ω ∈ Σ f (R) with card[n ∈ N : An ∈ Ω] = ℵ0 , Σ f (R) := [( n∈M An : M ∈ B f (N)) : (An : n ∈ N) ∈ Γ(R)]. Let φ(M) := ∪{An : n ∈ M}, M ⊂ B f (N), Σ be a filter of neighborhoods of 0/ in B f (N), then φ(Σ) ∈ Ψ f (R). In view of § 4.1.14 [Con84] we have µ(φ(Σ)) → 0 for each µ ∈ M(R, G, R′ ). We take a symmetric neighborhood V ′ ∋ 0 in G with 3V ′ ⊂ U. Then there are L ∈ H and M ∈ B f (N) such that µg (φ(M ′ )) − νh (φ(M”)) ∈ V ′ (or in addition D˜ Ub ∗ µg (φ(M ′ )) − D˜ Ub ∗ νh (φ(M”)) ∈ V ′ ) for each ([µ], [ν]) ∈ L¯ 2 ∩ U and M ′ , M” ∈ B(N \ M) (see Lemma 5.6). We choose U such that µg (φ(Mo )) − νh (φ(Mo )) ∈ V ′ (or in addition D˜ Ub ∗ µg (φ(Mo )) − D˜ Ub ∗ νh (φ(Mo )) ∈ V ′ ) for each ([µ], [ν]) ∈ U and Mo ⊂ M, consequently, µg (φ(M ′ )) − νh (φ(M”)) ∈ 3V ′ ⊂ U (or additionally D˜ Ub ∗ µg (φ(M ′ )) − D˜ Ub ∗ νh (φ(M”)) ∈ 3V ′ ) for M ∩ M ′ = M ∩ M”, the last statement follows from Corollary 5.5. ˆ 4 -filter in L (or P), τG ∋ U ∋ 0, 5.8. Corollary. Let (An : n ∈ N) ∈ (R), H be a Φ g then there are (L, M) ∈ H × B f (N) such that µ (∪(An : n ∈ M ′ )) ∈ U (or additionally S D˜ Ub ∗ µg ( n∈M′ An ) ∈ U) for [µ] ∈ L¯ and M ′ ∈ B(N \ M). S Proof. Let us take a fixed ν and W from § 5.7 with ν( n∈M′ An ) ∈ U˜ (or additionally S S ˜ where 2U˜ ⊂ U, 0 ∈ U˜ ∈ τG , g ∈ W . ˜ and (µg − ν)( n∈M′ An ) ∈ U), D˜ Ub ∗ ν( n∈M′ An ) ∈ U) ′ c 5.9. Theorem. Let U ⊂ R × τe × S =: Z be such that id : LU → L (or id : PU → P) be

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uniformly Φ4 -continuous, {H′ (A, R′ ) : A ∈ R} ⊂ Ψ f (R). Then ′ ′ ′ (a) L (or P) is Φ4 -closed in GR × RSo ×S (or GR × RSo ×S × GS ×K×R ), Mo is Φ4 -closed in GR ; (b) if G and R are Φi -compact, then LU (or PU ) is Φi -compact, where i ∈ (1, 2, 3, 4); (c) if G × Ro is sequentially complete, then LU (or PU ) is also; (d) if (0, 0) is the Gδ -subset in G × R, then LU ( or PU ) is Φ2 -compact; if additionally G × R is sequentially complete, then LU (or PU ) is Φ4 -compact; (e) LU (or PU ) is Hausdorff. Proof. We recall that a subset A ⊂ E of a topological or of a uniform space E is called Φi -closed (or compact), if for each Φi (A)-filter F is satisfied lim F ⊂ A (or lim F 6= 0/ respectively, that is, this definition of compactness differs from the usual). In view of Theorem 4.2.14[Con84] and Proposition 5.4 is accomplished 5.9(a). From §§ 2.1.14, 1.8.11[Con84] and Theorem 5.7, Φi -compactness of M and Mo , also from the completeness of Ro , L1 (K, v, C) it follows (b). Then (c) follows from (a) and § 1.8.7[Con84]; (d) follows from (c) and § 1.6.4[Con84], since G×R is Φ2 -compact. From M(R, G; R′ )U ′ (see §4.2.14[Con84]), RoS ×S and L1 (K, v, C) being Hausdorff it follows (d). ˆ 4 -filters on L (or P), also let φ : 5.10. Theorem. Let Φ be a set of Φ ′ L × R × S × S(∨ × K) → G × Ro (∨ × Y (v)) be such that (φ(µ, ρµ (∗, ∗)(∨ ηµ ), A, g, x) := (µ(A), ρµ (g, x)(∨ ηµg (t,U∗ , A))). If {H′ (A, R′ )|A ∈ R} ⊂ Ψ f (R), then a mapping φ : R × ′ M (R),G;R′ × RSo ×S (∨ × Y (v)) gives R′ -regular quasi-invariant measure S′ × S(∨ × K) → GΦ′o (or in addition a pseudo-differentiable measure) (A, g, x) → φ(∗, A, g, x) (or (A, g, x,t) → ′ φ(∗, A, g, x,t)), satisfying Conditions 5.1.(a, b), where Φ′ := πMo (Φ), in RSo ×S the topology corresponds to the topology in L. Proof follows from §§ 5.4 and 5.8. 5.11. Note. Let it be a sequence {[µn ] : n ∈ N} of quasi-invariant measures (or pseudodifferentiable measures) converging uniformly in uniformity 5.1.(ii) and fulfilling Conditions 5.1.(a,b) then in accordance with Corollary 5.8 {(µn )g : n ∈ N} (or also [D˜ Ub ∗ µgn : n]) is uniformly σ-additive for each fixed g ∈ S′ . Moreover, it is uniformly σ-additive by g ∈ W for each given B ∈ R such that gB ⊂ V for suitable open W in S′ and V in S. For L this means that for each 0 ∈ D ∈ τR and e ∈ U ∈ τG there are W ∋ e, d ∈ P”, c > 0, n and V ∋ e, a compact subset VU , e ∈ VU ⊂ V with µm (C) ∈ U (or D˜ Ub ∗ µm (C) ∈ U in addition for P) for C ∈ R and C ⊂ S \VU , with ρµm (g, x) − ρµ j (h, x′ ) ∈ D (or in addition D˜ Ub ∗ (µgm − µhj )(A) ∈ U for A ∈ B f (VU ) for P) whilst g, h ∈ W for each x, x′ ∈ VU with d(x, x′ ) < c and m, j > n. In view of Theorem 5.9 there exists limn→∞ (µn , ρµn (g, x)) = (y, d(y; g, x) ∈ L, that is, a quasiinvariant measure (or pseudo-differentiable measure limn [µ]n = [µ] ∈ P). Therefore, they are analogs of the Radon theorem for quasi-invariant measure and pseudo-differentiable measures.

1.6.

Measures with Particular Properties

1. Note. In [Lud00f, Lud99s] non-Archimedean polyhedral expansions of ultra-uniform spaces were investigated and the following theorem was proved (see also Appendix B).

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2. Theorem. Let X be a complete ultra-uniform space and K be a locally compact field. Then there exists an irreducible normal expansion of X into the limit of the inverse system S = {Pn , fnm , E} of uniform polyhedra over K, moreover, lim S is uniformly isomorphic with X, where E is an ordered set, fnm : Pm → Pn is a continuous mapping for each m ≥ n; particularly for the ultra-metric space (X, d) with the ultra-metric d the inverse system S is the inverse sequence. 3. Theorem. Let X be a complete separable ultra-uniform space and let K be a locally compact field. Then for each marked b ∈ C there exists a nontrivial measure µ on X which is a restriction of a measure ν in a measure space (Y, B f (Y ), ν) = lim{(Ym , B f (Ym ), νm ), f¯nm , E} on X and νm is quasi-invariant and pseudo-differentiable for b ∈ C relative to a dense subspace Y ′ m for each m, where Yn := c0 (K, αn ), f¯nm : Ym → Yn is a normal (that is, K-simplicial non-expanding) mapping for each m ≥ n ∈ E, f¯nm |Pm = fnm . Moreover, if X is not locally compact, then the family F of all such µ contains a subfamily G of pairwise orthogonal measures with the cardinality card(G ) = 2c , where c := card(R). Proof. Choose a polyhedral expansion of X in accordance with Theorem 2. Each mapping fnm is K-simplicial, that is, fnm is affine on each simplex s of a polyhedra Pm and fnm (s) is a simplex in Pn , also each fnm is non-expanding: ρPn ( fnm (x), fnm (y)) ≤ ρPm (x, y) for each x, y ∈ Pm . Since Yn is totally disconnected normed space and each simplex in Pm is the corresponding ball in Ym and each Pm is the uniform polyhedra, that is, (i) sup{diam(si : i)} < ∞, and (ii) inf{dist(si , s j ) : i 6= j} > 0, then fnm can be extended to a normal mapping f¯nm : Y m → Yn and such that Ym can be supplied with the corresponding uniform polyhedral structure (that is, partition into disjoint union of simplices satisfying Conditions (i, ii) above). Since X is separable and K is a locally compact field, then each space Yn is of countable type over K and E can be chosen countable. On each Xn take a probability measure νn such that νn (Xn \ Pn ) < εn , ∑n∈E εn < 1/5. In accordance with § 3.20.1 and § 4.2.1 each νn can be chosen quasiinvariant and pseudo-differentiable for b ∈ C relative to a dense K-linear subspace Y ′ n . Since E is countable and ordered and each Ym is supplied with the uniform polyhedral structure and the mapping f¯nm is normal for each m ≥ n, then a family νn can be chosen by transfinite induction consistent, that is, f¯nm (νm ) = νn for each m ≥ n in E, f¯nm (Y ′ m ) = Y ′ n . Then X = lim{Pn , fnm , E} ֒→ Y . Since f¯nm are K-linear, then ( f¯nm )−1 (B f (Yn )) ⊂ B f (Ym ) for S each m ≥ n ∈ E. Therefore, ν is correctly defined on the algebra n∈E fn−1 (B f (Yn )) of subsets of Y , where fn : X → Xn are K-linear continuous epimorphisms. Since ν is nonnegative and bounded by 1, then by the Kolmogorov theorem ν has an extension on the σ-algebra B f (Y ) and hence on its completion A f (Y, ν). Put Y ′ := lim{Y ′ m , f¯nm , E}. Then νm on Ym is quasi-invariant and pseudo-differentiable for b ∈ C relative to Y ′ m . From ∑n εn < 1/5 it follows, that 1 ≥ µ(X) ≥ ∏n (1 − εn ) > 1/2, hence µ is nontrivial. To prove the latter statement use the Kakutani theorem for ∏n Yn and then consider the embeddings X ֒→ Y ֒→ ∏n Yn such that projection and subsequent restriction of the measure ∏n νn on Y and X are nontrivial, which is possible due to the proof given above. If ∏n νn and ∏n ν′ n are orthogonal on ∏n Yn , then they give measures ν and ν′ which are orthogonal on X. 4. Notes. In [Lud00a] and in [BV97, Eva89, Eva91, Eva93, Koc95, Koc96, Lud0341,

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Sat94] analogs of a Gaussian measure and of a Wiener measure were considered. That construction is generalized below and additional properties are proved concerning moments of a Gaussian measure and an analog of the Itoˆ formula. Others constructions are discussed in comments (see §1.7). j Let X be a locally K-convex space equal to a projective limit lim{X j , φl , ϒ} of Banach spaces over a local field K such that X j = c0 (α j , K), where the latter space consists of vectors x = (xk : k ∈ α j ), xk ∈ K, kxk := supk |xk |K < ∞ and such that for each ε > 0 the set {k : |xk |K > ε} is finite, α j is a set, that is convenient to consider as an ordinal due to j Kuratowski-Zorn lemma [Eng86, Roo78]; ϒ is an ordered set, φl : X j → Xl is a K-linear j continuous mapping for each j ≥ l ∈ ϒ, φ j : X → X j is a projection on X j , φl ◦ φl = φ j for j j each j ≥ l ∈ ϒ, φlk ◦ φl = φk for each j ≥ l ≥ k in ϒ. Consider also a locally R-convex j space, that is a projective limit Y = lim{l2 (α j , R), ψl , ϒ}, where l2 (α j , R) is the real Hilbert space of the topological weight w(l2 (α j , R)) = card(α j )ℵ0 . Suppose B is a symmetric non-negative definite (bilinear) nonzero functional B : Y 2 → R. 5. Definitions and Notes. A measure µ = µq,B,γ on X with values in R is called a q-Gaussian measure, if its characteristic functional µˆ has the form µˆ (z) = exp[−B(vq (z), vq (z))]χγ (z) on a dense K-linear subspace Dq,B,X in X ∗ of all continuous K-linear functionals z : X → K of the form z(x) = z j (φ j (x)) for each x ∈ X with vq (z) ∈ DB,Y , where B is a nonnegative definite bilinear R-valued symmetric functional on a dense R-linear subspace DB,Y in Y ∗ , B : D2B,Y → R, j ∈ ϒ may depend on z, z j : X j → K is a continuous K-linear functional such that z j = ∑k∈α j ekj zk, j is a countable convergent series such that zk, j ∈ K, ekj is a continuous K-linear functional on X j such that ekj (el, j ) = δkl is the Kroneker delta symbol, el, j is the standard orthonormal (in the non-Archimedean sense) basis in c0 (α j , K), vq (z) = vq (z j ) := q/2 {|zk, j |K : k ∈ α j }. It is supposed that z is such that vq (z) ∈ l2 (α j , R), where q is a positive constant, χγ (z) : X → S1 is a continuous character such that χγ (z) = χ(z(γ)), γ ∈ X, χ : K → S1 is a character of K as an additive group (about a character see, for example, § VI.25 [HR79] and § III.1 [VVZ94]). If Y is a Hilbert space with a scalar product (∗, ∗), then due to the Riesz theorem there exists E ∈ L(Y ) such that B(y1 , y2 ) = (Ey1 , y2 ) for each y1 , y2 ∈ Y . A symmetric nonnegative definite operator E (or sometimes the corresponding B) is called a correlation operator of a measure µ. 6. Proposition. A q-Gaussian measure on X is σ-additive on some σ-algebra A of subsets of X. Moreover, a correlation operator B is of class L1 , that is, Tr(B) < ∞, if and only if each finite dimensional over K projection of µ is a σ-additive q-Gaussian Borel measure. Proof. From Definition 5 it follows, that each one dimensional over K projection µxK of a measure µ is σ-additive on the Borel σ-algebra B f (K), where 0 6= x = ek,l ∈ Xl . Therefore, µ is defined and finite additive on a cylindriS l −1 l cal algebra U := k1 ,...,kn ;l φ−1 l [(φk1 ,...,kn ) (B f (spanK {ek1 ,l , . . . , ekn ,l }))], where φk1 ,...,kn : Xl → spanK (ek1 ,l , . . . , ekn ,l ) is a projection on a K-linear span of vectors ek1 ,l , . . . , ekn ,l . This means that µ is a bounded quasi-measure on U. Since µˆ is the positive definite function, then µ is real-valued. In view of the non-Archimedean analog of the Bochner-

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Kolmogorov theorem (see § 2.27 above) µ has an extension to a σ-additive probability measure on a σ-algebra σU, that is, a minimal σ-algebra of subsets of X containing U. If J : X j → X j is a K-linear operator diagonal in the basis {ek, j : k}, then for z such that z(x) = z j (φ j (x)) for each x ∈ X and a symmetric non-negative definite operator F as in § 5 (i) F(vq (z ◦ J), vq (z ◦ J)) = E(vq (z), vq (z)), where (ii) Ek,l = Fk,l |Jk,k |q/2 |Jl,l |q/2 for each k, l ∈ α j . If F ∈ La (that is, F a ∈ L1 ) and J ∈ Lq (that is, diag(v1 (Jl,l ) : l) ∈ Lq ), then (iii) E ∈ Laq/(a+q) for each a > 0 (see Theorem 8.2.7 [Pie65]). In particular, taking a tending to ∞ and F = I we get E ∈ Lq , since L∞ is the space of bounded linear operators. Using the orthonormal bases in X j for each j we get the embedding of X j into its topologically dual space X j∗ of all continuous K-linear functionals on X j . For each z ∈ X ∗ there exists a non-Archimedean direct sum decomposition X = Xz ⊕ ker(z), where Xz is a one dimensional over K subspace in X. Therefore, the set Dq,B,X of functionals z on X from § 5 separates points of X. More generally consider in each X j a sequence of projection operators PVn, j on subspaces Vn, j := { i, j z : i = 1, . . . , n}, where { i, j z : i ∈ N} is the orthonormal basis in X j . Then consider J in this new basis and the transition operator from the standard basis to the new one. The composition of these two operators generates the corresponding operator C on Y which is in general non-diagonal. If for a given one dimensional over K subspace W in X it is the equality B(vq (z), vq (z)) = 0 for each z ∈ W , then the projection µW of µ is the atomic measure with one atom being a singleton. If B ∈ L1 , then B(vq (z), vq (z)) and hence µˆ (z) is correctly defined for each z ∈ Dq,B,X . It remains to establish that µ is σ-additive if and only if J ∈ Lq (c0 (ω0 , K)) and γ ∈ c0 (ω0 , K). We have µ j (K \ B(K, 0, r)) ≤ C ≤ C1

Z

Z

y∈R,|y|>r

x∈K,|x|>r

exp(−|x/ζ j |q )|ζ j |−1 v(dx)

exp(−|y|q |ζ j |−q )|ζ j |−1 dy,

where C > 0 and C1 > 0 are constants independent from ζ j for b0 > p3 and each r > b0 , 1 ≤ q < ∞ is fixed (see also the proof of Lemma 2.8 above and Theorem II.2.1 [DF91]). Evidently, g(γ) is correctly defined for each g ∈ c0 (ω0 , K)∗ if and only if γ ∈ c0 (ω0 , K). In this case the character χg(γ) : K → C is defined and χg(γ) = ∏∞j=1 χg j γ j . Due to Lemma 2.3 above, if J ∈ Lq (c0 ) and γ ∈ c0 (ω0 , K), then µ is σ-additive. Let 0 6= g ∈ c∗0 . Since K is the local field there exists x0 ∈ c0 such that |g(x0 )| = kgk and kx0 k = 1. Put g j := g(e j ). Then kgk ≤ sup j |g j |, since g(x) = ∑ j x j g j , where x = x j e j := ∑ j x j e j with x j ∈ K. Consequently, kgk = sup j |g j |. We enumerate the standard orthonormal basis {e j : j ∈ N} such that |g1 | = kgk. There exists an operator E on c0 with matrix elements Ei, j = δi, j for each i, j > 1, E1, j = g j for each j ∈ N. Then |det Pn EPn | = kgk for each n ∈ N, where Pn are the standard projectors on spanK {e1 , . . . , en } [LD02] (see also Appendix and comments to it). When g ∈ {e∗j : j ∈ ω0 }, then evidently, µg has the form given by Equation (iii), since µi (K) = 1 for each i ∈ ω0 , where e∗j (ei ) = δi, j for each i, j. Suppose now that J ∈ / Lq (c0 ). For this we consider µg (K \ B(K, 0, r)) ≥ R q −1 ∑ j x∈K,|x|>r C exp(−|x/ζ j | )|ζ j | v(dx), where g = (1, 1, 1, . . . ) ∈ c∗0 = l ∞ (ω0 , K). On the other hand, there exists a constant C2 > 0 such that for b0 > p3 and each r > b0 there is the

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following inequality: x∈K,|x|>r C exp(−|x/ζ j |q )|ζ j |−1 v(dx) ≥ R R −r C2 [ r∞ exp(−|y|q |ζ j |−q )|ζ j |−1 dy + −∞ exp(−|y|q |ζ j |−q )|ζ j |−1 dy]. From the estimates of Lemma II.1.1 [DF91] and using the substitution z = y1/2q for y > 0 and z = (−y)1/2q for y < 0 we get that µg is not σ-additive, consequently, µ is not σ-additive, since Pg−1 (A) are cylindrical Borel subsets for each A ∈ B f (K), where Pg z = g(z) is the induced projection on K for each z ∈ c0 . 7. Corollary. A q-Gaussian measure µ from Proposition 6 with Tr(B) < ∞ is quasiinvariant and pseudo-differentiable for some b ∈ C relative to a dense subspace Jµ ⊂ Mµ = {a ∈ X : vq (a) ∈ E 1/2 (Y )}. Moreover, if B is diagonal, then each one-dimensional projection µg has the following characteristic functional:     (i) µˆ g (h) = exp − ∑ β j |g j |q |h|q χg(γ) (h), R

j

where g = (g j : j ∈ ω0 ) ∈ c0 (ω0 , K)∗ . Proof. Using the projective limit reduce consideration to the Banach space X. The first statement follows from Theorems 3.12, 4.2 and 4.4 (see also [Lud00a]). To find Mµ consider a ∈ X and the q-Gaussian measures µ(dz) and µa (dz) := µ(−a+dz). Each Hellinger integral H(µa,n , µn ) has a value in [0, 1], hence ∏∞ n=1 H(µa,n , µn ) either diverges to zero and µa ⊥ µ or converges to a number 0 < β ≤ 1 and µa ∼ µ (see Theorem 3.3.1). Suppose that µa is not orthogonal to µ, then µa ∼ µ and there exists µa (dx)/µ(dx) = limn→∞ µVan (dxn )/µVn (dxn ) ∈ L1 (X, B , µ, C), where Vn := spanK (e1 , . . . , en ), n ∈ N, µVn is the projection of µ on Vn , xn := ( 1 x, . . . , n x), j x ∈ K for each j, xn ∈ V n . But µa (dx)/µ(dx) = lim [µa (dxn )/λn (dxn )][µVn (dxn )/λn (dxn )]−1 , n→∞

where λn is the Haar nonnegative measure, hence µVan (dxn )/λ(dxn ) ∈ L1 (Vn , B n , λn , C) for each n. Choose an orthonormal basis ( j z : j ∈ N) in X and an operator G : X → X such that G j z = j a j z, j a 6= 0 for each j, hence µ(G−1 dy) has the correlation operator CEC, where y ∈ G(X), G−1 : G(X) → X, C is defined by G and the transition operator from the standard orthonormal basis (e j : j) to ( j z : j) (see also § II.6.21). It is possible to take G such that CEC is the bounded continuous operator on Y and there exists the bounded continuous operator (CEC)−1 on Y . The Fourier operator is unitary on L2 (X, B , µ, C). Therefore, the existence of µˆ a relative to the measure µ implies vq (a) ∈ E 1/2 (Y ). Since vq (ξa) = vq (ξ)vq (a) and vq (a j + b j ) ≤ max(vq (a j ), vq (b j )) for each ξ ∈ K and a, b ∈ X, a = ∑ j a j e j , then the family of all such a ∈ X with vq (a) ∈ E 1/2 (Y ) is the K-linear subspace. Vice versa, if vq (a) ∈ E 1/2 (Y ), then the proof above shows that there exists µˆ a relative to µ and hence there exists the limit limn→∞ [µVan (dxn )/λn (dxn )][µVn (dxn )/λn (dxn )]−1 = µa (dx)/µ(dx). For the verification of Formula (i) it is sufficient at first to consider the measure µ on the algebra UP of cylindrical subsets in c0 . Then for each projection µg , where g ∈ spanK (e1 , . . . , em )∗ , we have: µˆ g (h) =

Z

··· K

Z

K

χe (hz)µ1 (dx1 ) · · · µm (dxm ),

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where e = (1, . . . , 1) ∈ Qnp , h ∈ K, n := dimQp K, xi ∈ Kei , z = g(x), x = (x1 , . . . , xm ), i consequently, µˆ g (h) = ∏m ˆ i (hgi ), since χe (hg(x)) = ∏m i=1 µ i=1 χe (hi gi x ) for each x ∈ spanK (e1 , . . . , em ). Let J be a subspace of all x ∈ X such that vq (x) ∈ B1/2 (DB,Y ). Since DB,Y is R-linear and vq (az) = |a|q/2 vq (z) for each a ∈ K, vq (y + z) ≤ max(vq (y), vq (z)), then J is K-linear and µ is quasi-invariant relative to J. In view ofRthe Parceval-Steklov equality and definition of pseudo-differentiabilty and convergence of K exp(−a(vq (x))2 )|x|m v(dx) < ∞ for each m ≥ 0 and each a > 0, where v is a nonnegative nontrivial Haar measure on K it follows, that µ is pseudo-differentiable for each b ∈ C with Re(b) > 0. Since B ∈ Lq , then µ is the Radon measure, consequently, the continuation of µ from UP produces µ on the Borel σ-algebra of c0 , hence limm→∞ µˆ Qm g (h) = µˆ g (h), where Qm is the natural projection on spanK (e1 , . . . , em )∗ for each m ∈ N such that Qm (g) = (g1 , . . . , gm ). Using expressions of µˆ i we get Formula (i). From this it follows, that if B ∈ Lq , then µˆ (g) exists for each g ∈ c∗0 if and only if γ ∈ c0 , since µˆ g (h) = µˆ (gh) for each h ∈ K and g ∈ c∗0 . 8. Corollary. Let X be a complete locally K-convex space of separable type over a local field K, then for each constant q > 0 there exists a nondegenerate symmetric positive definite operator B ∈ L1 such that a q-Gaussian measure is σ-additive on B f (X) and each its one dimensional over K projection is absolutely continuous relative to the nonnegative Haar measure on K. Proof. A space Y from § 4 corresponding to X is a separable locally R-convex space. Therefore, Y in a weak topology is isomorphic with Rℵ0 from which the existence of B follows. For each K-linear finite dimensional over K subspace S a projection µS of µ on S ⊂ X exists and its density µS (dx)/w(dx) relative to the non-negative nondegenerate Haar measure w on S is the inverse Fourier transform F −1 (ˆµ|S∗ ) of the restriction of µˆ on S∗ (see about the Fourier transform on non-Archimedean spaces § VII [VVZ94]). For such B each one dimensional projection of µ corresponding to µˆ has a density that is a continuous function belonging to L1 (K, w, B f (K), R), where w denotes the nonnegative Haar measure on K. 9. Proposition. Let µq,B,γ and µq,E,δ be two q-Gaussian measures with correlation operators B and E of class L1 , then there exists a convolution of these measures µq,B,γ ∗ µq,E,δ , which is a q-Gaussian measure µq,B+E,γ+δ . Proof. Since B and E are nonnegative, then (B + E)(y, y) = B(y, y) + E(y, y) ≥ 0 for each y ∈ Y , that is, B + E is nonnegative. Evidently, B + E is symmetric. In view of [Pie65] B + E is of class L1 . Therefore, µq,B+E,γ+δ is the σ-additive q-Gaussian measure together with µq,B,γ and µq,E,δ in accordance with Proposition 6. Moreover, µq,B+E,γ+δ is defined on the σ-algebra σUB+E containing the union of σ-algebras σUB and σUE on which µq,B,γ and µq,E,δ are defined correspondingly, since ker(B + E) ⊂ ker(B) ∩ ker(E). Since µˆ q,B+E,γ+δ = µˆ q,B,γ µˆ q,E,δ , then µq,B+E,γ+δ = µq,B,γ ∗ µq,E,δ (see Proposition A.12 in Appendix and use projective limits). 9.1. Remark and Definition. A measurable space (Ω, F) with a probability real-valued measure λ on a covering σ-algebra F of a set Ω is called a probability space and it is denoted by (Ω, F, λ). The random variable ξ induces a normalized measure νξ (A) := λ(ξ−1 (A)) in X and a new probability space (X, B, νξ ). Let T be a set with a covering σ-algebra R and a measure η : R → R. Denote by Lq (T, R , η, H) the completion of the set of all R -step functions f : T → H relative to the

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following norm: R q (1) k f kη,q := ( t∈T k f (t)kH η(dt))1/q for 1 ≤ q < ∞ and (2) k f kη,∞ := ess − supη,t∈T k f (t)kH , where H is a Banach space over K. For 0 < q < 1 this is the metricR space with the metric q (3) ρq ( f , g) := ( t∈T k f (t) − g(t)kH η(dt))1/q . If H is a complete locally K-convex space, then H is a projective limit of Banach spaces H = lim{Hα , παβ , ϒ}, where ϒ is a directed set, παβ : Hα → Hβ is a K-linear continuous mapping for each α ≥ β, πα : H → Hα is a K-linear continuous mapping such that παβ ◦ πα = πβ for each α ≥ β (see § 6.205 [NB85]). Each norm pα on Hα induces a pre-norm p˜α on H. If f : T → H, then πα ◦ f =: fα : T → Hα . In this case Lq (T, R , η, H) is defined as a completion of a family Rof all step functions f : T → H relative to the family of pre-norms (1′ ) k f kη,q,α := ( t∈T p˜α ( f (t))q η(dt))1/q , α ∈ ϒ, for 1 ≤ q < ∞ and (2′ ) k f kη,∞,α := ess − supη,t∈T p˜α ( f (t)), α ∈ ϒ, or pseudo-metrics R (3′ ) ρq,α ( f , g) := ( t∈T [ p˜α ( f (t) − g(t))]q η(dt))1/q , α ∈ ϒ, for 0 < q < 1. Therefore, Lq (T, R , η, H) is isomorphic with the projective limit lim{Lq (T, R , η, Hα ), παβ , ϒ}. For example, T may be a subset of R. If T ⊂ F with a nonArchimedean field F, then we can consider the non-Archimedean time parameter also. If T is a zero-dimensional T1 -space, then denote by Cb0 (T, H) the Banach space of all continuous bounded functions f : T → H supplied with the norm: (4) k f kC0 := supt∈T k f (t)kH < ∞. For a set T and a complete locally K-convex space H over K consider the product K-convex space H T := ∏t∈T Ht in the product topology, where Ht := H for each t ∈ T . Then take on either X := X(T, H) = Lq (T, R , η, H) or X := X(T, H) = Cb0 (T, H) or on X = X(T, H) = H T a covering σ-algebra B such that B ⊃ B f (X). Consider a random variable ξ : ω 7→ ξ(t, ω) with values in (X, B), where t ∈ T . Consider T such that card(T ) > n. For X = Cb0 (T, H) or X = H T define X(T, H; (t1 , . . . ,tn ); (z1 , . . . , zn )) as a closed sub-manifold in X of all f : T → H, f ∈ X such that f (t1 ) = z1 , . . . , f (tn ) = zn , where t1 , . . . ,tn are pairwise distinct points in T and z1 , . . . , zn are points in H. For X = Lq (T, R , η, H) and pairwise distinct points t1 , . . . ,tn in T with supp(η) ⊃ {t1 , . . . ,tn } define X(T, H; (t1 , . . . ,tn ); (z1 , . . . , zn )) as a closed sub-manifold which is the completion relative to the norm k f kη,q of a family of R -step functions f : T → H such that f (t1 ) = z1 , . . . , f (tn ) = zn . In these cases X(T, H; (t1 , . . . ,tn ); (0, . . . , 0)) is the proper K-linear subspace of X(T, H) such that X(T, H) is isomorphic with X(T, H; (t1 , . . . ,tn ); (0, . . . , 0)) ⊕ H n , since if f ∈ X, then f (t) − f (t1 ) =: g(t) ∈ X(T, H;t1 ; 0) (in the third case we use that T ∈ R and hence there exists the embedding H ֒→ X). For n = 1 and t0 ∈ T and z1 = 0 we denote X0 := X0 (T, H) := X(T, H;t0 ; 0). 9.2. Definitions. We define a (non-Archimedean) stochastic process w(t, ω) with values in H as a random variable such that: (i) the differences w(t4 , ω) − w(t3 , ω) and w(t2 , ω) − w(t1 , ω) are independent for each chosen (t1 ,t2 ) and (t3 ,t4 ) with t1 6= t2 , t3 6= t4 , such that either t1 or t2 is not in the twoelement set {t3 ,t4 }, where ω ∈ Ω; (ii) the random variable ω(t, ω) − ω(u, ω) has a distribution µFt,u , where µ is a probability real-valued measure on (X(T, H), B) from § 9.1, µg (A) := µ(g−1 (A)) for g : X → H such that g−1 (R H ) ⊂ B and each A ∈ R H , Ft,u (w) := w(t, ω)−w(u, ω) for each w ∈ Lq (Ω, F, λ; X),

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−1 where 1 ≤ q ≤ ∞, R H is a covering σ-algebra of H such that Ft,u (R H ) ⊂ B for each t 6= u in T ; (iii) we also put w(0, ω) = 0, that is, we consider a K-linear subspace Lq (Ω, F, / X0 is the closed subspace of X as in § 9.1. λ; X0 ) of Lq (Ω, F, λ; X), where Ω 6= 0, 10. Definitions. Let B and q be as in § 6 and denote by µq,B,γ,a the corresponding q-Gaussian measure on H. Let ξ be a stochastic process with a real time t ∈ T ⊂ R (see Definition 9.2), then it is called a non-Archimedean q-Wiener process with real time, if (ii)′ the random variable ξ(t, ω)−ξ(u, ω) has a distribution µq,(t−u)B,γ for each t 6= u ∈ T . Let ξ be a stochastic process with a non-Archimedean time t ∈ T ⊂ F, where F is a local field, then ξ is called a non-Archimedean q-Wiener process with F-time, if (ii)” the random variable ξ(t, ω) − ξ(u, ω) has a distribution µq,ln[χF (t−u)]B,γ for each t 6= u ∈ T , where χF : F → S1 is a continuous character of F as the additive group. 11. Proposition. For each given q-Gaussian measure a non-Archimedean q-Wiener process with real (F respectively) time exists. Proof. In view of Proposition 9 for each t > u > b a random variable ξ(t, ω) − ξ(b, ω) has a distribution µq,(t−b)B,γ for real time parameter. If t, u, b are pairwise different points in F, then ξ(t, ω) − ξ(b, ω) has a distribution µq,ln[χF (t−b)]B,γ , since ln[χF (t − u)] + ln[χF (u − (q)

b)] = ln[χF (t − b)]. This induces the Markov quasi-measure µx0 ,τ on (∏t∈T (Ht , Ut )), where Ht = H and Ut = B f (H) for each t ∈ T (see § VI.1.1 [DF91] and § 3 in [Lud0321]). Therefore, the Chapman-Kolmogorov equation is accomplished: P(b, x,t, A) =

Z

P(b, x, u, dy)P(u, y,t, A) H

for each A ∈ B f (H). An abstract probability space (Ω, F, λ) exists due to the Kolmogorov theorem, hence the corresponding space Lr exists. Therefore, conditions of Definitions 10 are satisfied (see also 4.1 [Lud0321]). 12. Proposition. Let ξ be a q-Gaussian process with values in a Banach space H = c0 (α, K) a time parameter t ∈ T and a positive definite correlation operator B of trace class and γ = 0, where card(α) ≤ ℵ0 , either T ⊂ R or T ⊂ F. Then either (i) (ii)

lim Mt k(vq (e1 (ξ(t, ω)), . . . , vq (eN (ξ(t, ω)))k2l2 = tTr(B) or

N∈α

lim Mt k(vq (e1 (ξ(t, ω)), . . . , vq (eN (ξ(t, ω))k2l2 = [ln(χF (t))]Tr(B) respectively.

N∈α

Proof. At first we consider moments of a q-Gaussian measure µq,B,γ . We define moR q j1 j k ments mk (e , . . . , e ) := H v2q (e j1 (x)) · · · v2q (e jk (x))µq,B,γ (dx) for linear continuous functionals e j1 , . . . , e jk on H such that el (e j ) = δlj , where in our previous notation {e j : j ∈ α} is the standard orthonormal base in H. Consider partial pseudo-differential operators P ∂uj given by the equation (iii)

u P ∂ j ψ(x)

u ˆ x))(x), := F−1 ˜ j (|x˜ j |K ψ(

where the norm |b|K = modK (b) on K is chosen coinciding with the modular function associated with the nonnegative nondegenerate Haar measure w on K (about the modular

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81

ˆ := F j (ψ) is the Fourier transform of ψ by a varifunction see [Wei73]), u ∈ C \ {−1}, ψ able x j ∈ K such that F j is defined relative to the Haar measure w on K [VVZ94]. From R R the change of variables formula K f (ax + b)g(x)w(dx) = K f (y)g((y − b)/a)|a|−1 K w(dy) for each f and g ∈ L2 (K, B f (K), w, C), a 6= 0 and b ∈ K, also the Fubini theorem and the Fourier transform on K it follows that f−α ∗ fu+1 = fu+1−α for u 6= α and ΓK (u + 1)|ξ j |−u−1 = F(|x j |uR), where fu (x j ) := |x j |u−1 /ΓK (u), ΓK is the non-Archimedean gamma K 1 function, ΓK (u) := K |z|u−1 K χ(z)w(dz), χ : K → S is the character of K as the additive group such that χ(z) := ∏mj=1 χ p (z′ j ), z′ j ∈ Qp , z = (z′ 1 , . . . , z′ m ) ∈ K for K considered as the Qp -linear space, m ∈ N, dimQp K = m, χ p : Qp → S1 is the standard character such that χ p (y) := exp(2πi{y} p ), {y} p := ∑l 1 and {y} p = 0 for |y|Qp ≤ 1, y = ∑l al pl , al ∈ {0, 1, . . . , p − 1}, l ∈ Z, min(l : al 6= 0) =: ord p (y) > −∞. Therefore, u n n−u Γ (n)/Γ (n − u), where n ∈ C \ {−1}. A function ψ for which ∂u ψ P ∂ j |x j | = |x j | K K P j exists is called pseudo-differentiable of order u by variable x j . q/2 From mk (e j1 , . . . , e jk ) = F−1 (|x j1 |q/2 · · · |x jk |q/2 F(µ))(0), since F(hg) = F(h) ∗ F(g) for q/2 functions h and g in the Hilbert space L2 (K, B f (K), w, C) it follows that m2k (e j1 , . . . , e j2k ) = q/2 q/2 ˆ (0) = ([ P Dq/2 ]2k µˆ (0)).(e j1 , . . . , e j2k ), where P Dq/2 is a K-linear pseudoP ∂ j1 · · · P ∂ j2k µ differential operator by x ∈ H such that q/2

(P Dq/2 ψ(x)).e j := P ∂ j ψ(x). Then q/2 (iv) m2n (e j1 , . . . , e j2n ) = (−1)n (n!)−1 [ P Dq/2 ]2n [B(vq (z), vq (z)]n .(e j1 , . . . , e j2n ) = (n!)−1 ∑σ∈Σ2n Bσ( j1 ),σ( j2 ) · · · Bσ( j2n−1 ),σ( j2n ) , since γ = 0 and χγ (z) = 1, where Σk is the symmetric group of all bijective mappings σ of the set {1, . . . , k} onto itself, Bl, j := B(e j , eRl ), since Y ∗ = Y for Y = l2 (α, R). Therefore, for each B ∈ L1 and A ∈ L∞ we have H A(vq (x), vq (x))µq,B,0 (dx) = q/2 limN∈α ∑Nj=1 ∑Nk=1 A j,k m2 (e j , ek ) = Tr(AB). In particular for A = I and µq,tB,0 corresponding to the transition measure of ξ(t, ω) we get Formula (i) for a real time parameter, using µq,ln[χF (t)]B,0 we get Formula (ii) for a time parameter belonging to F, since ξ(t0 , ω) = 0 for each ω. 13. Corollary. Let H = K and ξ, B = 1, γ be as in Proposition 12, then   Z b Z q φ(t, ω)dt φ(t, ω)|dξ(t, ω)|K = M (i) M t∈[a,b]

a

for each a < b ∈ T with real time, where φ(t, ω) ∈ L2 (Ω, U, λ,C00 (T, R)) ξ ∈ Lq (Ω, U, λ, X0 (T, K)), (Ω, U, λ) is a probability measure space. R q Proof. Since t∈[a,b] φ(t, ω)|dξ(t, ω)|K q = limmax j (t j+1 −t j )→0 ∑Nj=1 φ(t j , ω)|ξ(t j+1 , ω) − ξ(t j , ω)|K for λ-almost all ω ∈ Ω, then applyq ing Formula 12.(i) to each |ξ(t j+1 , ω) − ξ(t j , ω)|K and taking the limit by finite partitions a = t1 < t2 < · · · < tN+1 = b of the segment [a, b] we get Formula 13.(i). 14. Remarks. In the classical case withRq = 2 and R instead of K there is analoR gous formula M([ t∈[a,b] φ(t, ω)dBt (ω)]2 ) = M[ ab φ(t, ω)2 dt] known as the Itoˆ formula (see the classical case in [Bou63-69, DF91]). Another analogs of the Itoˆ formula were given

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Sergey V. Ludkovsky

in [Lud0341] (see also comments in § 1.7). Certainly it is impossible to get in the nonArchimedean case all the same properties of Gaussian measures and Wiener process (Brownian motion) as in the classical case. Therefore, there are different possibilities for seeking non-Archimedean analogs of Gaussian measures and Wiener processes depending on a set of properties supplied with these objects. Giving our definitions we had the intention to take into account the most important properties. Since F(χγ )(y) = δ(y − γ) and [δ(y − γ) ∗ h(y)](x) = h(x − γ) for any continuous function R R q/2 q/2 h, then H |x j1 − γ j1 | · · · |x jk − γ jk | dµq,B,γ = H |x j1 |q/2 · · · |x jk |q/2 dµq,B,0 , consequently, γ plays in some sense the mean value role. If A > 0 on Y = l2 (α, K), then µq,B,0 {x : A(vq (x), vq (x)) ≥ 1} ≤ Tr(AB) and µq,B,0 {x : |A(vq (x), vq (x)) − Tr(AB)| ≤ c(Tr(AB))1/2 } ≥ 1 − 2kABk/c2 for each c > 0 due to the Chebyshev inequality and Formula 12.(iv). 15. Definitions and Notes. Consider a pseudo-differential operator on H = c0 (α, K) such that (i) A = (−i)k bkj1 ,..., jk P ∂ j1 · · · P ∂ jk , ∑ 0≤k∈Z; j1 ,..., jk ∈α

where ∈ R, P ∂ jk := P ∂1jk . If there exists n := max{k : bkj1 ,..., jk 6= 0, j1 , . . . , jk ∈ α}, then n is called an order of A, Ord(A). If A = 0, then by definition Ord(A) = 0. If there is not any such finite n, then Ord(A) = ∞. We suppose that the corresponding form A˜ on L k k Y is continuous into C, where bkj1 ,..., jk

˜ (ii) A(y) =−

∑

(−i)k bkj1 ,..., jk y j1 · · · y jk ,

0≤k∈Z; j1 ,..., jk ∈α

˜ y ∈ l2 (α, R) =: Y . If A(y) > 0 for each y 6= 0 in Y , then A is called strictly elliptic pseudodifferential operator. The phase multiplier (−i)k is inserted into the definition of A for in the definition of P ∂ j it was omitted in comparison with the classical case. Let X be a complete locally K-convex space, let Z be a complete locally Cconvex space. For 0 ≤ n ∈ R a space of all functions f : X → Z such that f (x) and ( P Dk f (x)).(y1 , . . . , yl(k) ) are continuous functions on X for each y1 , . . . , yl(k) ∈ {e1 , e2 , e3 , . . . } ⊂ X ∗ , l(k) := [k] + sign{k} for each k ∈ N such that k ≤ [n] and also for k = n is denoted by P C n (X, Z) and f ∈ P C n (X, Z) is called n times continuously pseudodifferentiable, where [n] ≤ n is an integer part of n, 1 > {n} := n − [n] ≥ 0 is a fractional part of n, sign(b) = 1 for each b > 0, sign(0) = 0, sign(b) = −1 for b < 0. Then T∞ ∞ n P C (X, Z) := n=1 P C (X, Z) denotes a space of all infinitely pseudo-differentiable functions. 16. Theorem. Let A be a strictly elliptic pseudo-differential operator on H = c0 (α, K), card(α) ≤ ℵ0 , and let t ∈ T = [0, b] ⊂ R. Suppose also that u0 (x − y) ∈ L2 (H, B f (H), µt A˜ , C) for each marked y ∈ H as a function by x ∈ H, u0 (x) ∈ Ord(A) (H, C). Then the non-Archimedean analog of the Cauchy problem PC (i) ∂u(t, x)/∂t = Au,

u(0, x) = u0 (x)

has a solution given by (ii) u(t, x) =

Z

H

u0 (x − y)µt A˜ (dy),

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83

where µt A˜ is a σ-additive Borel measure on H with a characteristic functional µˆ t A˜ (z) := ˜ 2 (z))]. exp[−t A(v Proof. In accordance with § § 4 and 15 we have Y = l2 (α, R). In view of the condi˜ 2 (z))] is continuous on H ֒→ H ∗ for each t ∈ R tions of this theorem the function exp[−t A(v such that the family H of continuous K-linear functionals on H separates points in H. In view of the Minlos-Sazonov theorem 2.35 above it defines a σ-additive Borel measure on H for each t > 0 and hence for each t ∈ (0, b]. The functional A˜ on each ball of radius ˜ L [span (e ,...,e )]k , when n tends to 0 < R < ∞ in Y is a uniform limit of its restrictions A| n K 1 k L k the infinity, since A˜ is continuous on k Y . Since u0 (x − y) ∈ L2 (H, B f (H), µt A˜ , C) and a space of cylindrical functions is dense in the latter Hilbert space, then due to the ParcevalSteklov equality and the Fubini theorem it follows that limP→I FPx u0 (Px))ˆµt A˜ (y + Px) converges in L2 (H, B f (H), µt A˜ , C) for each t, since µt1 A˜ ∗ µt2 A˜ = µ(t1 +t2 )A˜ for each t1 , t2 and t1 + t2 ∈ T , where P is a projection on a finite dimensional over K subspace HP := P(H) in H, HP ֒→ H, P tends to the unit operator I in the strong operator topology, FPx u0 (Px) denotes a Fourier transform by the variable Px ∈ HP . Consider a function v := Fx (u), ˜ 2 (x))v(t, x), consequently, v(t, x) = v0 (x) exp[−t A(v ˜ 2 (x))]. From then ∂v(t, x)/∂t = −A(v −1 u(t, x) = Fx (v(t, x)), where as above Fx (u) denotes the Fourier transform by the variable xR ∈ H such that Fx (u(t, x)) = limn→∞ Fx1 ,...,xn u(t, x). Therefore, u(t, x) = u0 (x) ∗ F−1 µt A˜ ) = x (ˆ 2 (H, B f (H), µ , C) and µ is the bounded measure u (x − y)µ (dy), since u (x − y) ∈ L 0 t A˜ R t A˜ t A˜ H 0 R on B f (H) and | H u0 (x − y)µt A˜ (dy)| ≤ ( H |u0 (x − y)|2 µt A˜ (dy))µt A˜ (H) < ∞. 17. Note. In the particular case of Ord(A) = 2 and A˜ corresponding to the Laplace ˜ operator, that is, A(y) = ∑l, j gl, j yl y j , Equation 12.(i) is (the non-Archimedean analog of) the heat equation on H. This provides the interpretation of the 2-Gaussian measure µt A˜ = µ2,t A,0 ˜ . For dimK H < ∞ the density µt A˜ (dx)/w(dx) is called the heat kernel, where w is the nonnegative nondegenerate Haar measure on H. For Ord(A) < ∞ the form A˜ 0 (y) corresponding to sum of terms with k = Ord(A) in Formula 15.(ii) is called the principal symbol of operator A. If A˜ 0 (y) > 0 for each y 6= 0, then A is called an elliptic pseudo-differential operator. Evidently, Theorem 16 is true for ˜ 2 (z))] is the bounded continuous real-valued elliptic A of Ord(A) < ∞, since exp[−t A(v positive definite function. 18. Remark and Definitions. Let linear spaces X over K and Y over R be as in § 4 and B be a symmetric nonnegative definite (bilinear) operator on a dense R-linear subspace DB,Y in Y ∗ . A quasi-measure µ with a characteristic functional µˆ (ζ, x) := exp[−ζB(vq (z), vq (z))]χγ (z) for a parameter ζ ∈ C with Re(ζ) ≥ 0 defined on Dq,B,X is called a complex-valued Gaussian measure and is denoted by µq,ζB,γ also, where Dq,B,X := {z ∈ X ∗ : there exists j ∈ ϒ such that z(x) = z j (φ j (x)) ∀x ∈ X, vq (z) ∈ DB,Y }. 19. RProposition. Let X = Dq,B,X and B be positive definite, then for each function f (z) := X χz (x)ν(dx) with a complex-valued measure ν of finite variation and each Re(ζ) > 0 there exists Z Z (P) f (Pz)µζB (dz) (i) f (z)µζB (dz) = lim X

=

Z

X

P→I X

exp(−ζB(vq (z), vq (z)))χγ (z)ν(dz),

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Sergey V. Ludkovsky

where µ(P) (P−1 (A)) := µ(P−1 (A)) for each A ∈ B f (XP ), P : X → XP is a projection on a K-linear subspace XP , a convergence P → I is considered relative to a strong operator topology. Proof. A complex-valued measure ν can be presented as ν = ν1 − ν2 + iν3 − iν4 , where ν j are nonnegative measures, j = 1, . . . , 4, i = (−1)1/2 . Using the projective limit decomposition of X and § 2.27 above we get that Z

(ii)

X

f (z)µζB (dz) = lim

Z

P→I X

(P)

f (Pz)µζB (dz).

On the other hand, for each finite dimensional over K subspace XP (iii)

Z

X

(P) f (Pz)µζB (dz)

=

Z

XP

{exp(−ζB(vq (z), vq (z)))χγ (z)}|XP νXP (dz).

Since each measure ν j is non-negative and finite, then due to Lemma 2.3 and § 2.5 above there exists the limit lim

Z

P→I XP

=

{exp[−ζB(vq (z), vq (z))]χγ (z)}|XP νXP (dz)

Z

X

exp(−ζB(vq (z), vq (z)))χγ (z)ν(dz).

20. Proposition. If conditions of Proposition 19 are satisfied and (i)

Z

XP

| f (Px)|wXP (dx) < ∞

for each finite dimensional over K subspace XP in X, then Formula 19.(i) is accomplished for ζ with Re(ζ) = 0, where wXP is a non-negative nondegenerate Haar measure on XP . Proof. The finite dimensional over K distribution P µXq,iB,γ /wXP (dx) = F−1 (ˆµq,iB,γ |XP )

is locally wXP -integrable, but does not belong to the space L1 (XP , B f (XP ), wXP , C). In view of Condition 20.(i) above and the Fubini theorem and using the Fourier transform of generalized functions (see § VII.3 [VVZ94]) we get Formulas 19.(ii, iii). Taking the limit by P → I we get Formula 19.(i) in the sense of distributions. 21. Remark. A measure µq,iB,γ is the non-Archimedean analog of the Feynman quasimeasure. Put Z Z (i) F f (x)µq,iB,γ (dx) := lim f (x)µq,ζB,γ (dx) ζ→i X

X

if such limit exists. If conditions of Proposition 19 are satisfied, then ψ(ζ) := X f (x)µq,ζB,γ (dx) is the holomorphic function on {ζ ∈ C : Re(ζ) > 0} and it is continuous on {ζ ∈ C : Re(ζ) ≥ 0}, consequently,

R

(ii)

F

Z

X

f (x)µq,iB,γ (dx) =

Z

X

exp{−iB(vq (x), vq (x))}χγ (x)ν(dx).

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Above were defined non-Archimedean analogs of Gaussian measures with specific properties, but usual Gaussian measures does not exist on non-Archimedean spaces as the following theorem shows. 22. Theorem. Let X be a Banach space of separable type over a locally compact field K. If on B f (X) there exists a nontrivial real-valued (probability) usual Gaussian measure, then K = R. Proof. Let µ be a nontrivial usual Gaussian real-valued measure on B f (X). Then by the definition its characteristic functional µˆ must be positive definite complex-valued function such that µˆ (0) = 1, lim|y|→∞ µˆ (y) = 0 for each y ∈ X ∗ \ {0}, where X ∗ is the topological conjugate space to X of all continuous K-linear functionals f : X → K. Moreover, there exist a K-bilinear functional g and a compact non-degenerate K-linear operator T : X ∗ → X ∗ with ker(T ) = {0} and a marked vector x0 ∈ X such that µˆ x0 (y) = f (g(Ty, Ty)) for each y ∈ X ∗ , where µx0 (dx) := µ(−x0 + dx), x ∈ X. Since K is locally compact, then X ∗ is nontrivial and separates points of X (see [NB85, Roo78]). Each one-dimensional over K projection of a Gaussian measure is a Gaussian measure and products of Gaussian measures are Gaussian measures, hence convolutions of Gaussian measures are also Gaussian measures. Therefore, µˆ x0 : X ∗ → C is a nontrivial character: µˆ x0 (y1 + y2 ) = µˆ x0 (y1 )ˆµx0 (y2 ) for each y1 and y2 in X ∗ . If char(K) = 0 and K is a non-Archimedean field, then there exists a prime n number p such that Qp is the subfield of K. Then µˆ (pn y) = (ˆµ(y)) p for each n ∈ Z and y ∈ X ∗ \ {0}, particularly, for n ∈ N tending to the infinity we have limn→∞ pn y = 0 and n limn→∞ µˆ x0 (pn y) = 1, limn→∞ (ˆµx0 (y)) p = 0, since limn→∞ µˆ x0 (p−n y) = 0 and |ˆµx0 (y)| < 1 for y 6= 0. This gives the contradiction, hence K can not be a non-Archimedean field of zero characteristic. Suppose that K is a non-Archimedean field of characteristic char(K) = p > 0, then K is isomorphic with the field of formal power series in variable t over a finite field Fp . Therefore, µˆ x0 (py) = 1, but µˆ x0 (y) p 6= 1 for y 6= 0, since limn→∞ µˆ x0 (t −n y) = 0. This contradicts the fact that µˆ x0 need to be the nontrivial character, consequently, K can not be a non-Archimedean field of nonzero characteristic as well. It remains the classical case of X over R or C, but the latter case reduces to X over R with the help of the isomorphism of C as the R-linear space with R2 . 23. Theorem. Let µq,B,γ and µq,B,δ be two q-Gaussian measures. Then µq,B,γ is equivalent to µq,B,δ or µq,B,γ ⊥ µq,B,δ according to vq (γ− δ) ∈ B1/2 (DB,Y ) or not. The measure µq,B,γ is orthogonal to µg,B,δ , when q 6= g. Two measures µq,B,γ and µg,A,δ are either equivalent or orthogonal. 24. Theorem. The measures µq,B,γ and µq,A,γ are equivalent if and only if there exists a positive definite bounded invertible operator T such that A = B1/2 T B1/2 and T − I ∈ L2 (Y ∗ ). Proof. Using the projective limit reduce consideration to the Banach space X. Then proof of theorems 23, 24 follows from the consideration of characteristic functionals of measures, the Kakutani theorem 3.3.1 and the fact that the Fourier transform F is the unitary operator on L2 (K, B f (K), v, C) due to the Parceval-Steklov equality, where v denotes the Haar normalized nonnegative measure on K. Therefore, it is possible to proceed with the characteristic functionals µˆ q,B,δ and µˆ g,A,γ instead of measures. If g 6= q then the measure c c )/(µ µq,B,γ is orthogonal to µg,B,δ , since limR>0,R+n→∞ (µq,B,γ )Xn (XR,n g,B,δ )Xn (XR,n ) = 0, c := X \ B(X , 0, R), X := span (e : m = n, n + 1, . . . , 2n), for each q > g, where XR,n n n n K m (µq,B,γ )Xn is the projection of the measure µq,B,γ on Xn . Each Hellinger integral in the Kaku-

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tani theorem is in [0, 1] ⊂ R, consequently, the product in the Kakutani theorem is either converging to a positive constant or diverges to zero, hence two measures µq,B,γ and µg,A,δ are either equivalent or orthogonal (see also the classical case in § II.3 [Kuo75]).

1.7.

Comments

Real-valued Haar measures on locally compact topological groups are described in details, for example, in [Bou63-69, HR79, FD88]. S.N. Evans had used stochastic approach to study non-Archimedean analogs of Gaussian measures [Eva89, Eva91]. 1. Definition. Consider a normed space (Y, k ∗ kY ) over a non-Archimedean field F. A subset X in Y is called orthogonal if for each finite subset {x1 , . . . , xn } ⊂ X and each a1 , . . . , an is satisfied the equality: k ∑nj=1 a j x j kY = max1≤ j≤n ka j kkx j kY . An orthogonal subset X is called orthonormal, if kxkY = 1 for each x ∈ X. 2. Definition. Let (E, E ) be a measurable vector space over a local field F. A random variable ξ ∈ L2 (E, P, R) is called a Gaussian random variable (is distributed by a Gaussian probability measure) if for each two independent copies ξ1 and ξ2 of ξ and each orthonormal vectors (a1 , a2 ) and (b1 , b2 ) ∈ F2 the pair (ξ1 , ξ2 ) has the same distribution as (a1 ξ1 + a2 ξ2 , b1 ξ1 + b2 ξ2 ). 3. Theorem. A random variable ξ ∈ L2 (E, P, R) that is not almost surely zero is Gaussian if and only if its distribution is a cutoff of the Haar measure λ: P({ξ ∈ dx}) = π−n ψ(π−n kxk)λ(dx) for some n ∈ Z, where ψ is the characteristic function of [0, 1] ⊂ R, |π| < 1 is the generator of the normalization group ΓF , E is locally compact. 4. Theorem. Suppose (E, k ∗ kE ) is a separable Banach space with topological dual space E ∗ and P is a F-Gaussian measure on E. Put S := {x ∈ E : |T (x)| ≤ kT (X)k∞ ∀T ∈ E ∗ }. Then (i) the group S is the closed support of P; (ii) the group S is compact; (iii) if M is a measurable vector subspace of E, then P(M) is either 1 or 0, depending on whether S ⊂ M or not. 5. Remark. A random variable ξ ∈ L2 (E, P, R) was called also a random variable with values in E in terminology of S.N. Evans (see also [DF91, Eva88, Eva89, Eva91]). Gaussian random fields with values in Qp and controlled by real-valued measures were also constructed in [AK91] with the help of forward and backward Kolmogorov equations. The property of invariance of the Gaussian distribution under orthogonal non-Archimedean transformations leads to a measure with compact support and equivalent to the Haar measure up to a constant multiplier. Together with results above this gives another proof of the fact that in the non-Archimedean case there does not exist a measure having all the same properties as the Gaussian measure in a real Banach space. On the other hand, it shows that the orthogonality condition being too strong in the non-Archimedean case leads to a measure, which is a restriction of the Haar measure on a compact subset in the case of locally compact E. Theorem 4 shows, that S.N. Evans has considered measures with compact support also in a non-locally compact F-linear space E. This means, that such measures of Theorems 3, 4 are not quasi-invariant, but in the classical case Gaussian measures are quasi-invariant.

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V.S. Vladimirov (see [Vla89, VVZ94] and references therein) and A.N. Kochubei [Koc95] also have considered integrals over Qp of the function f (x) := χ p (ax2 + bx) with parameters a, b ∈ Qp as analogs of Gaussian integrals and they evaluated them, where χ p is the complex valued character of Qp as the additive group. Certainly, because of |χ p (z)| = 1 for each z ∈ Qp , the function χ p (ax2 + bx) has not characteristic graph of the Gaussian distribution tending to zero while |x| tends to the infinity. All this terminology Gaussian integrals and Gaussian measures is very conditional and optional, since in the times of K.F. Gauss no any non-Archimedean fields were studied and non-Archimedean analysis was not existent, because non-Archimedean fields had begun to be investigated only in the end of the 19-th century and the non-Archimedean normalization comes back from the Ostrowski’s theorem (see [Roo78, Wei73]). Any use of suitable nonArchimedean measures or integrals depends on concrete problems. Pseudo-differential operators considered in this chapter were first studied by V.S. Vladimirov [Vla89] and also were used in non-Archimedean quantum mechanics [VV89, VVZ94]. They were used for solutions of the non-Archimedean analog of the heat equation (see,R for example, [Koc96]). It was proved [VVZ94], for example, for p 6= 2 and a 6= 0, that B(Qp ,0,pN ) χ p (ax2 + bx)dx =: I(p, N, a, b) has values I(p, N, a, b) = pN ω(pN |b| p ), when |a| p p2N ≤ 1, −1/2

I(p, N, a, b) = λ p (a)|a| p

χ p (−b2 /(4a))ω(p−N |b/a| p ),
when |a| p p2N > 1, where λ p (a) = 1 for even ν, λ p (a) = ap0 , if ν is odd and p =
1 (mod 4), λ p (a) = i ap0 , if ν is odd and p = 3 (mod 4), where ν is such that
a = pν (a0 + a1 p + · · · ), ν ∈ Z, a j ∈ {0, 1, . . . , p − 1}, a ∈ Qp , a0 6= 0, np is the Legen
dre symbol for each prime p and n an integer prime to p, such that np = 1 for n being a
quadratic residue modulo p, and np = −1 otherwise, ω(y) = 1 for 0 ≤ y ≤ 1, ω(y) = 0 for y > 1, dx denotes the Haar nonnegative measure on Qp . Taking the limit while N tends to infinity gives Z

−1/2

Qp

χ p (ax2 + bx)dx := lim I(p, N, a, b) = λ p (a)|a| p N→∞

χ p (−b2 /(4a)).

For a local field K and the character χ of rank zero of K as the additive group and a radial function f (x) := g(kxk) it was also evaluated the integral Z

∞

K

χ(xξ) f (x)dx = (1 − q−1 )kξk−1 ∑ q−n g(q−n kξk−1 ) − kξk−1 g(qkξk−1 ) n=0

for each ξ 6= 0, where K is the finite algebraic extension of Qp and q := p f , f ≥ 1 is the index of inertia, e f = (K : Qp ), e ≥ 1 is the ramification index, B(K, 0, 1)/{x ∈ K : |x| < 1} = Fq is the finite field consisting of q elements. There is a generalization of the Minlos theorem on locally K-convex spaces. In the case of K = R it states, that there exists a bijective correspondence between continuous positive definite functions on a nuclear locally convex topological vector space and Radon probability measures on the Borel σ-algebra of the topological conjugate space E ∗ supplied with the weak topology σ(E ∗ , E) provided by the Fourier transform. For properties of

88

Sergey V. Ludkovsky

measures to be quasi-invariant or pseudo-differentiable it does not play any role. This theorem only permits to get an information about a measure by its characteristic functional, that is also very important. Ma¸drecki has proved its non-Archimedean analog in [Mad91c]. 6. Theorem. Let E be a Hausdorff locally K-convex space, where K is a local field. Then any continuous positive definite function on E is the Fourier transform of a Radon probability measure on the Borel σ-field B f (E ∗ ) of the topological conjugate space E ∗ supplied with the ∗-weak topology σ(E ∗ , E). Conversely, if E is barreled, then the Fourier transform of a Radon probability measure on B f (E ∗ ) is a continuous positive definite function on E. 7. Remark. In [Sat94] an analog of a Wiener measure in a Banach space with a measurable norm was studied by T. Satoh. It was further development of some results of S.N. Evans. 8. Definitions. A countable subset {ei : i ∈ N} is called an orthogonal Schauder base in a normed space H over a non-Archimedean field K if it satisfies the following two conditions: (1) for each v ∈ H there is the unique sequence {ci : ci ∈ K, i ∈ N} such that v = ∑i ci ei ; (2) for any converging series ∑i ci ei , we have | ∑∞ i=1 ci ei | = maxi |ci ei |. If in addition to the above conditions |ei | = 1 for each i, then {ei : i} is called an orthonormal Schauder basis. A K-linear map P ∈ L(H) is called an orthogonal projection, if P2 = P and Im(P) := P(H) is orthogonal to Ker(P) := P−1 (0). A probability real-valued measure ν on K is said to be admissible, if: (1) the measure ν is isometry invariant absolute continuous with respect to the Haar measure µ and (2) the value of the Radon-Nykodim derivative dν/dµ(x) at x = πn is a non-decreasing function of n, where |π| < 1 is generator of the normalization group ΓK . The cylinder measure Gν with parameter ν is the function on Cyl(H) defined by the following formula: Gν (P−1 (F)) := νP(H) (F), where Cyl(H) denotes the algebra of all cylinder subsets defined with the help of orthogonal projections on finite dimensional over K subspaces in H, dimK P(H) < ∞, P−1 (F) ∈ Cyl(H), F ∈ B f (P(H)). A semi-norm k ∗ k in H is called measurable if for every ε > 0 there exists P ∈ FOP(H) satisfying kxk ≤ εkxk for each x ∈ Ker(P), where FOP(H) denotes the family of all orthogonal projection operators with finite-dimensional ranges over K. Suppose B is a completion of H relative to the measurable norm k ∗ k in H and B∗ be the topological dual space of all continuous K-linear functionals f : B → K. 9. Proposition. The measure Gν is not σ-additive. 10. Definitions. Let Cyl ∗ (B) be the family of all cylinder subsets in B of the form: T := {x ∈ B : (P1 (x), . . . , Pn (x)) ∈ E}, where E ∈ A f (Kn , µn ), P1 , . . . , Pn ∈ Cyl(H). The Wiener measure Wν with parameter ν is defined on Cyl ∗ (B) of the form: Wν (T ) := Gν (T ∩ H) for each T ∈ Cyl ∗ (H). 11. Theorem. The Wiener measure Wν extends to a σ-additive measure on the sigma algebra σ(Cyl ∗ (B)) generated by Cyl ∗ (B). 12. Remark. Relations between topologies and Borel structures are given, for example, in [Chr74, Kur66] and references therein. 13. Definitions. By a Borel structure B (X) on a set X there is understood a system (a particular subfamily) of subsets of X which is closed with respect to the operations of

Real-Valued Measures

89

complement and countable union, and which is nonempty. Thus B (X) is the paving of X, it is also called a σ-field. A (Borel) measurable mapping f of (X, B (X)) into (Y, B (Y )) is a mapping f : X → Y such that f −1 (B) ∈ B (X) for each B ∈ B (Y ). A Borel isomorphism is a bijective mapping f of X onto Y such that both f and f −1 are (Borel) measurable. A measurable space (X, B (X)) is called separated, if for all x 6= y ∈ X there exists A ∈ B (X) such that x ∈ A, but y ∈ / A. A measurable space (X, B (X)) is called separable, if there exists a sequence {An ∈ B (X) : n ∈ N} which generates B (X). It is called countably separated, if there exists a separable subfield which is separated. 14. Theorem. A measurable space (X, B (X)) is Borel isomorphic with a subset of the real segment [0, 1] equipped with the subspace Borel structure (induced from the Borel structure generated by the usual topology on [0, 1]) if and only if it is separable and separated. 15. Definitions. A Polish space X is a Hausdorff topological space which can be equipped with a metric d generating its topology and (X, d) is complete. An analytic topological space Y is a Hausdorff space which is the continuous image of a Polish space X. If the Hausdorff topological space Y is an injective continuous image of a Polish space, it is called a standard topological space. 16. Theorem. If X is an analytic topological space, then there exists a surjective continuous mapping f from Nℵ0 onto X, where Nℵ0 is supplied with the product topology. 17. Definitions. Let (X, τX ) be a topological space with a Hausdorff topology τY , then A ⊂ X is called of the first category (in X), if A is a countable union of closed nowhere dense subsets in X. Subsets A of X which can not be so represented are called of the second category. The BP-field is the σ-field of subsets A of X for which there exists U ∈ τX such that A △ U := (A \ U) ∪ (U \ A) is of the first category. A Hausdorff topological space is called a Baire space, if every its open subset is of the second category. A topological group G is called analytic, if G is analytic as a topological space. A topological group G is called σ-bounded, if G can be covered with countably many left translations of every neighborhood. 18. Theorem. Let G be a topological group and A be a BP-measurable subset of the second category. Then A ◦ A−1 is a neighborhood. 19. Theorem. Any BP-measurable homomorphism f from a topological group G which is of the second category in itself to a σ-bounded topological group H is continuous. If both G and H are analytic and G is of the second category, then each homomorphism f : G → H with analytic graph is continuous. 20. Theorem. Let (G, τG ) be a Baire topological group with a quasi-invariant nontrivial nonnegative measure µ relative to a dense Baire subgroup (G′ , τG′ ) (in itself) with a BP-measurable quasi-invariance factor ρµ (h, g) : G′ × G → R, then ρµ (h, g) is continuous in (h, g) ∈ G′ × G. Proof. It follows from Theorems 18, 19 using the co-cycle property of a quasiinvariance factor ρµ (hv, g) = ρµ (v, h−1 g)ρµ (h, g) on a topological group G relative to the left action of a dense subgroup G′ , where ρµ (h, g) := µ(h−1 dg)/µ(dg), µ is a measure on G. Let ′ B be a Borel subset in R, then ρ−1 µ (B) =: A is a BP-measurable subset in (G × G, τG′ × τG ).

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Sergey V. Ludkovsky

In particular, take B open in R. In view of the co-cycle condition and Definition 17 there exists V of the second category in G′ × G such that V is BP-measurable and A △ V is of the first category. Since (G′ × G, τG′ × τG ) is the Baire topological group, then A ◦ A−1 is a neighborhood of the unit element in G′ × G. 21. Note. The latter theorem shows, that the supposition of continuity of the quasiinvariance factor of this chapter is not very restrictive. Moreover, it is implied also under milder conditions. In details relations between Borel measurability and continuity of functions φ : (G′ × X) ∋ (h, x) 7→ hx := φ(h, x) ∈ X satisfying conditions φ(e, x) = x, φ(v, φ(h, x)) = φ(vh, x) for each v, h ∈ G′ and each x ∈ X were given in [Fid00], where X is a Polish topological space and G′ is a Polish topological group.

Chapter 2

Non-Archimedean Valued Measures 2.1.

Introduction

This Chapter is the continuation of the first one and treats the case of measures with values in non-Archimedean fields of zero characteristic, for example, the field Qp of p-adic numbers. There are specific features with formulations of definitions and theorems and their proofs, because of differences in the notions of σ-additivity of real-valued and Qp -valued measures, differences in the notions of spaces of integrable functions, quasi-invariance and pseudo-differentiability. For the s-free group G a measure m with values in a nonArchimedean field Ks satisfy Condition I.1.(H) only for an algebra of clopen (closed and open) subsets A, where a field Ks is a finite algebraic extension of Qs . Indeed, in the last case if a measure is locally finite and σ-additive on the Borel algebra of G, then it is purely atomic with atoms being singletons, so it can not be invariant relative to the entire Borel algebra (see Chapters 7-9 [Roo78]). The Lebesque convergence theorem has quite another meaning, the Radon-Nikodym theorem in its classical form is not applicable to the considered here case. A lot of definitions and theorems given below are the non-Archimedean analogs of results for real-valued measures of Chapter I. Frequently their formulations and proofs differ strongly. If proofs differ slightly from the case of real-valued measures of Chapter I, only general circumstances are given in for non-Archimedean-valued measures. In § 2 sequences of weak distributions, characteristic functions of measures and their properties are defined and investigated. The non-Archimedean analogs of the MinlosSazonov and Bochner-Kolmogorov theorems are given. Quasi-measures also are considered. In § 3 products of measures are considered together with their density functions. The non-Archimedean analog of the Kakutani theorem is investigated. In the present chapter broad classes of quasi-invariant measures are defined and constructed. Theorems about quasi-invariance of measures under definite linear and non-linear transformations U : X → X are proved. § 4 contains a notion of pseudo-differentiability of measures. This is necessary, because for functions f : K → Qs with s 6= p there is not any notion of differentiability (there is not such non-linear non-trivial f ), where K is a field such that K ⊃ Qp . There are given criteria for the pseudo-differentiability. In § 5 there are given theorems about convergence of measures with taking into account their quasi-invariance and pseudodifferentiability, that is, in the corresponding spaces of measures. The main results are Theorems 2.21, 2.30, 3.5, 3.6, 3.15, 3.19, 3.20, 4.2, 4.3, 4.5, 5.7-5.10.

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Sergey V. Ludkovsky

In this chapter notations of Chapter I are used also. Notations. Henceforth, K denotes a locally compact infinite field with a non-trivial norm, then the Banach space X is over K. In the present chapter measures on X have values in the field Ks , where Ks is a non-Archimedean field complete relative to its uniformity and such that Qs ⊂ Ks , where Qs is the s-adic field with the certain prime number s. In all theorems of this chapter and Chapter IV measures can be realized, when Ks is a local field, that is, a finite algebraic extension of Qs , but also they are true, when Ks is a broader field, for example, Cs ⊂ Ks or Us ⊂ Ks (see below). Henceforth, Cs denotes the uniform completion of the union of all local fields Ks with the multiplicative ultra-norm extending that of Qs . Let Us be a field obtained from Cs with the help of procedures of ultra-products and spherical completion such that its normalization group ΓUs = (0, ∞) (see [Dia84, Esc95] and references therein and comments below). We assume that K is s-free as the additive group, for example, either char(K) = 0, K is a finite algebraic extension of the field of p-adic numbers Qp or char(K) = p and K is isomorphic with a field Fp (θ) of formal power series consisting of elements x = ∑ j a j θ j , where a j ∈ Fp , |θ| = p−1 , Fp is a finite field of p elements, p is a prime number and p 6= s. These imply that K has the Haar measures with values in Ks [Roo78]. If X is a Hausdorff topological space with a small inductive dimension ind(X) = 0, then E denotes an algebra of subsets of X, as a rule E ⊃ Bco(X) for Ks -valued measures, where Bco(X) denotes an algebra of clopen (closed and open) subsets of X, B f (X) is a Borel σ-field of X, A f (X, µ) is the completion of E by a measure µ in § 2.1.

2.2.

Non-Archimedean Valued Distributions

2.1. For a Hausdorff topological space X with a small inductive dimension ind(X) = 0 [Eng86], henceforth, measures µ are given on a measurable space (X, E), where E is an algebra such that E ⊃ Bco(X), Bco(X) is an algebra of closed and at the same time open (clopen) subsets in X. We recall that a mapping µ : E → Ks for an algebra E of subsets of X is called a measure, if the following conditions are accomplished: / = 0, (i) µ is additive and µ(0) (ii) for each A ∈ E there exists the following norm kAkµ := sup{|µ(B)|Ks : B ⊂ A, B ∈ E} < ∞, (iii) if there is a shrinking family F, that is, for each / then limA∈F µ(A) = 0 (see A, B ∈ F there exist F ∋ C ⊂ (A ∩ B) and ∩{A : A ∈ F} = 0, also Chapter 7 [Roo78] and also about the completion A f (X, µ) of the algebra E by the measure µ). A measure with values in Ks is called a probability measure if kXkµ = 1 and µ(X) = 1. For functions f : X → Ks and φ : X → [0, ∞) there are used notations k f kφ := supx∈X (| f (x)|φ(x)), Nµ (x) := inf(kUkµ : U ∈ Bco(X), x ∈ X). Tight measures (that is, measures defined on E ⊃ Bco(X)) compose the Banach space M(X) with a norm kµk := kXkµ . Everywhere below there are considered measures with kXkµ < ∞ for µ with values in Ks , if it is not specified another.

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93

A measure µ on E is called Radon, if for each ε > 0 there exists a compact subset C ⊂ X such that kµ|(X\C) k < ε. Henceforth, M(X) denotes the space of norm-bounded measures, Mt (X) is its subspace of Radon norm-bounded measures. 2.1.1. Definition. Suppose that S is a subfamily of a covering ring R of X such that for each A and B in S there exists C ∈ S with C ⊂ A ∩ B, then S is called shrinking. For a function f : R → K or f : R → R the notation limA∈S f (A) = 0 means that for each ε > 0 there exists B ∈ S such that | f (A)| ≤ ε for each A ∈ S with A ⊂ B. 2.1.2. Notes. Put k f kµ := k f kNµ . Then for each A ⊂ X the function kAkµ := supx∈A Nµ (x) is defined such that its restriction on R coincides with that of given by Equation 2.1.(ii) (see also Chapter 7 [Roo78]). A R -step function f is a function f : X → K such that it is a finite linear combination over K of characteristic functions ChU of U ∈ R . A function f is called µ-integrable if there exists a sequence { fn : n ∈ N} of step functions such that limn→∞ k f − fn kNµ = 0. The Banach space of µ-integrable functions is denoted by L(µ) := L(X, R , µ, K). There exists a ring R µ of subsets A in X for which ChA ∈ L(µ). The ring R µ is the extension of the ring R such that R µ ⊃ R . For example, if K is locally compact, then the normalization group ΓK := {|x| : x ∈ K, x 6= 0} is discrete in (0, ∞) ⊂ R. If µ is a measure such that 0 < kµk < ∞, then there exists a ∈ K such that |a| = kµk−1 , since kµk ∈ ΓK for discrete ΓK , hence aµ is also the measure with kµk = 1. If kµk = 1, then µ is the nonzero measure. For such µ with µ(X) =: bX ∈ K if bX 6= 1 we can take a non-void new set Y and define on X0 := Y ∪ X a minimal ring R 0 / {Y }} and R 0 ⊃ R ∪ {Y }. Since kµk = 1, generated by R and {Y }, that is, R 0 ∩ Y = {0, then |bX | ≤ 1. Put µ(Y ) := 1 − bX , then there exists the extension of µ from R on R 0 such that kµk = 1 and µ(X0 ) = 1, since |1 − bX | ≤ max(1, |bX |) = 1. In particular, we can take a singleton Y = {y}. Therefore, probability measures are rather naturally related with nonzero bounded measures. This also shows that from kµk = 1 in general does not follow µ(X) = 1. Evidently, from µ(X) = 1 in general does not follow kµk = 1, for example, X = {0, 1}, / {0}, {1}, X}, µ({0}) = a, µ({1}) = 1 − a, where |a| > 1, hence kµk = |a| > 1. R = {0, Consider a non-void topological space X. A topological space is called zerodimensional if it has a base of its topology consisting of clopen subsets. A topological space X is called a T0 -space if for each two distinct points x and y in X there exists an open subset U in X such that either x ∈ U and y ∈ X \U or y ∈ U and x ∈ X \U. A covering ring R of a space X defines on it a base of zero-dimensional topology τR such that each element of R is considered as a clopen subset in X. If π : X → Y is a mapping such that π−1 (R Y ) ⊂ R X , then a measure µ on (X, R X ) induces a measure ν := π(µ) on (Y, R Y ) such that ν(A) = µ(π−1 (A)) for each A ∈ R Y . 2.2. If A ∈ Bco(L), then PL−1 (A) is called a cylindrical subset in X with a base A, BL := PL−1 (Bco(L)), B0 := ∪(BL : L ⊂ X, L is a Banach subspace , dimK X < ℵ0 ) (see §I.2.2). Let an increasing sequence of Banach subspaces Ln ⊂ Ln+1 ⊂ · · · such that cl(∪[Ln : n]) = X, dimK Ln = κn for each n be chosen, where cl(A) = A¯ denotes a closure of A in X for A ⊂ X. We fix a family of projections PLLnm : Lm → Ln such that PLLnm PLLkn = PLLkm for each m ≥ n ≥ k. A projection of the measure µ onto L denoted by µL (A) := µ(PL−1 (A)) for each A ∈ Bco(L) compose the consistent family: µLn (A) = µLm (PL−1 (A) ∩ Lm ) n

(1)

for each m ≥ n, since there are projectors PLLnm , where κn ≤ ℵ0 and there may be chosen

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κn < ℵ0 for each n. An arbitrary family of measures {µLn : n ∈ N} having property (1) is called a sequence of weak distributions (see also [DF91, Sko74]). 2.3. Lemma. A sequence of weak distributions {µLn : n} is generated by some measure µ on Bco(X) if and only if for each c > 0 there exists b > 0 such that kLn \ B(X, 0, r)kµLn ≤ c for each n ∈ N and supn kLn kµLn < ∞ for µ with values in Ks , where r ≥ b. Proof. For µ with values in Ks the necessity is evident. Recall Theorem 7.6 [Roo78]: (Ri). If µ is a measure on R , then Nµ is R -upper semi-continuous and for each A ∈ R µ and b > 0 the set {x ∈ A : Nµ (x) ≥ b} is R µ -compact (hence R -compact); (Rii). Conversely, let µ : R → K be additive. Assume that there exists an R -upper semi-continuous function φ : X → [0, ∞) such that |µ(A)| ≤ supx∈A φ(x) for each A ∈ R and {x ∈ A : φ(x) ≥ b} is compact for every b > 0. Then µ is a measure and Nµ ≤ φ. To prove the sufficiency it remains only to verify property (2.1.iii), since then kXkµ = supn kLn kµLn < ∞. Let B(n) ∈ E(Ln ), A(n) = PL−1 (B(n)), by the cited above Theorem n 7.6 [Roo78] for each c > 0 there is a compact subset C(n) ⊂ B(n) such that kB(n) \ C(n)kµLn < c, where kB(n) \ D(n)kµ ≤ max(kB(m) \ C(m)kµL(m) : m = 1, . . . , n) < c and T D(n) := nm=1 PL(m) −1 (C(m)) ∩ Ln ), PL−1 (E(Ln ) ⊂ E = E(X). If A(n) ⊃ A(n + 1) ⊃ · · · and n T T ′ ′ / where A′ (n) := PL−1 / then A (n + 1) ⊂ A (n) and n A′ (n) = 0, (D(n)), hence n A(n) = 0, n ′ kA(n)kµ ≤ kA (n)kµ + c. There may be taken B(n) as closed subsets in X. Remind the Hahn-Banach Theorem 4.8 [Roo78]: let E and F be normed spaces, D a linear subspace of E; assume that either D or F is spherically complete; then every S ∈ ¯ = kSk. L(D, F) has an extension S¯ ∈ L(E, F) such that kSk In accordance with the Alaouglu-Bourbaki theorem (see Exer. 9.202(a.3) [NB85]) if K is a locally compact field with a non-Archimedean multiplicative norm, X is a locally K-convex space and U is a neighborhood of zero in X, then its polar U o := { f ∈ X ′ : supx∈U | f (x)| ≤ 1} is σ(X ′ , X)-compact. In view of the Alaoglu-Bourbaki theorem and the Hahn-Banach theorem sets A(n) and B(X, 0, r) are weakly compact in X, hence, for each r > 0 there exists n with / Therefore, kA(n)kµ = kB(n)kµLn ≤ kLn \ B(X, 0, r)kµLn ≤ c and there B(X, 0, r) ∩ A(n) = 0. exists limn→∞ µ(A(n)) = 0, since c is arbitrary. 2.4. Definition and notations. A function φ : X → Ks of the form φ(x) = φS (PS x) is called a cylindrical function if φS is a E(S)-measurable function on a finite-dimensional over K space S in X. For φS ∈ L(S, µS , Ks ) := L(µS ) for µ with values in Ks we may define an integral by a sequence of weak distributions {µS(n) }: Z

X

φ(x)µ∗ (dx) :=

Z

φS(n) (x)µS(n) (dx),

where L(µ) is the Banach space of classes of µ-integrable functions ( f = g µ-almost everywhere, that is, kAkµ = 0, A := {x : f (x) 6= g(x)} is µ-negligible) with the following norm k f k := kgkNµ . 2.5. Remarks and definitions. In the notation of § I.2.6 all continuous characters χ : K → Cs have the form −1 χξ (x) = εz η((ξ,x)) (1)

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for each η((ξ, x)) 6= 0, χξ (x) := 1 for η((ξ, x)) = 0, where ε = 1z is a root of unity, z = pord(η((ξ,x))) , π j : K → R, η(x) := {x} p and ξ ∈ Qnp ∗ = Qnp for char(K) = 0, η(x) := π−1 (x)/p and ξ ∈ K∗ = K for char(K) = p > 0, x ∈ K, (see also § 25 [HR79]). Each χ is locally constant, hence χ : K → Ts is also continuous, where T denotes the discrete group of all roots of 1, Ts denotes its subgroup of elements with orders that are not degrees sm of s, m ∈ N. For a measure µ with values in Ks there exists a characteristic functional (that is, called the Fourier-Stieltjes transformation) θ = θµ : C(X, K) → Us : θ( f ) :=

Z

X

χe ( f (x))µ(dx),

(2)

where Ks ∪Cs ⊂ Us , either e = (1, . . . , 1) ∈ Qnp for char(K) = 0 or e = 1 ∈ K∗ for char(K) = p > 0, x ∈ X, f is in the space C(X, K) of continuous functions from X into K, in particular for z = f in the topologically conjugated space X ∗ over K, z : X → K, z ∈ X ∗ , θ(z) =: µˆ (z). It has the following properties: θ(0) = 1 for µ(X) = 1

(3a)

sup |θ( f )| = 1 for probability measures ;

(3b)

and θ( f ) is bounded on C(X, K); f

θ(z) is weakly continuous, that is, (X ∗ , σ(X ∗ , X))-continuous,

(4)

σ(X ∗ , X) denotes a weak topology on X ∗ , induced by the Banach space X over K. To each x ∈ X there corresponds a continuous linear functional x∗ : X ∗ → K, x∗ (z) := z(x), moreover, θ( f ) is uniformly continuous relative to the norm on Cb (X, K) := { f ∈ C(X, K) : k f k := sup | f (x)|K < ∞}. x∈X

Recall the non-Archimedean analog of the Lebesgue theorem (see Exer. 7.F [Roo78]) for µ with values in Ks . Let µ be a measure on R , let g ∈ L(µ) and let { f j : j} be a net of µ-integrable functions from X into Ks converging to a function f uniformly on R µ -compact subsets and such thatR | f j | ≤ |g| for every j. Then f ∈ L(µ) and lim j k f j − f kµ = 0 and R lim j X f j (x)µ(dx) = X f (x)µ(dx). Remind also the equicontinuity theorem: let X be a topological vector space and let X ′ be its topologically dual; a subset H of X ′ is equicontinuous if and only if H is contained in a polar V o of some neighborhood V of 0 in X (see (9.5.4) and Exer. 9.202 [NB85]). Property (4) follows from Lemma 2.3, boundedness and continuity of χe and the fact that due to the Hahn-Banach theorem there is xz ∈ X with z(xz ) = 1 for z 6= 0 such that z|(X⊖L) = 0 and Z Z θ(z) =

X

χe (PL (x))µ(dx) =

L

χe (y)µL (dy),

where L = Kxz , also due to the Lebesgue theorem. Indeed, for each c > 0 there exists a compact subset S ⊂ X such that kX \ Skµ < c, each bounded subset A ⊂ X ∗ is uniformly equicontinuous on S, that is, {χe (z(x)) : z ∈ A} is

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the uniformly equicontinuous family (by x ∈ S). On the other hand, χe ( f (x)) is uniformly equicontinuous on a bounded A ⊂ Cb (X, K) by x ∈ S. We call a functional θ finite-dimensionally concentrated, if there exists L ⊂ X, dimK L < ℵ0 , such that θ|(X\L) = µ(X). For each c > 0 and δ > 0 in view of Theorem 7.6[Roo78] recalled above there exists a finite-dimensional over K subspace L and compact S ⊂ Lδ such that kX \ Skµ < c. Let θL (z) := θ(PL z). This definition is correct, since L ⊂ X, X has the isometrical embedding into X ∗ as the normed space associated with the fixed basis of X, such that functionals z ∈ X separate points in X. If z ∈ L, then |θ(z) − θL (z)| ≤ c × b × q, where b = kXkµ , q is independent of c and b. Each characteristic functional θL (z) is Runiformly continuous by z ∈ L relative to the norm k ∗ k on L, since |θL (z) − θL (y)| ≤ | S′ ∩L [χe (z(x)) − χe (y(x))] R µL (dx)| +| L\S′ [χe (z(x)) − χe (y(x))] µL (dx)|, where the second term does not exceed 2C′ for kL\S′ kµL < c′ for a suitable compact subset S′ ⊂ X and χe (z(x)) is an uniformly equicontinuous by x ∈ S′ family relative to z ∈ B(L, 0, 1). For a field K denote by TK the group of all those roots of unity of K whose orders are not divisible by the characteristic p of the residue class field k of K. A K-valued character of a point-wise torsional group G is a continuous homomorphism G → TK . The K-valued characters form a group Gˆ.K . Remind the basic theorem about the Fourier-Stieltjes transform (see also Theorem 9.20 [Roo78]): let G be a torsional group compatible with Ks , then the Fourier-Stieltjes transform is an isomorphism of Banach algebras M(G) ≃ BUC(Gˆ.Ks ). A group G is supplied with a subgroup topology, if there exists a family U of its T subgroups so that H∈U H = {1} and the co-sets of H ∈ U form a subbase for a zerodimensional Hausdorff topology on G that renders G a topological group. Remind also that a topological group G with a subgroup topology is called torsional if every compact subset of G is contained in a compact subgroup of G. Therefore, θ(z) = lim θn (z) (5) n→∞

for each finite-dimensional over K subspace L, where θn (z) is uniformly equicontinuous and S finite-dimensionally concentrated on Ln ⊂ X, z ∈ X, cl( n Ln ) = X, Ln ⊂ Ln+1 for every n, for each c > 0 there are n and q > 0 such that |θ(z) − θ j (z)| ≤ cbq for z ∈ L j and j > n, q = const > 0 is independent of j, c and b. Let {e j : j ∈ N} be the standard orthonormal basis in X, e j = (0, . . . , 0, 1, 0, . . . ) with 1 in j-th place. Using Property 2.1.(iii) of µ, local constantness of χe , considering all z = be j and b ∈ K, we get that θ(z) on X is non-trivial, whilst µ is a non-zero measure, since due to Lemma 2.3 µ is characterized uniquely by {µLn : n}. Indeed, for µ with values in Ks a measure µV on V , dimKV < ℵ0 , this follows from the basic theorem about the Fourier-Stieltjes transform, where F(g)(z) := lim

Z

r→∞ B(V,0,r)

χe (z(x))g(x)m(dx),

z ∈ V, g ∈ L(V, µV , Us ), m is the Haar measure on V with values in Ks . Therefore, the mapping µ 7→ θµ is injective. 2.6. Theorem.Let µ1 and µ2 be measures in M(X) on the same algebra E, where Bco(X) ⊂ E ⊂ B f (X) such that µˆ 1 ( f ) = µˆ 2 ( f ) for each f ∈ Γ. Then µ1 = µ2 , where

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X = c0 (α, K), α ≤ ω0 , Γ is a vector subspace in a space of continuous functions f : X → K separating points in X. Proof. Let at first α < ω0 , then due to § 2.5 µ1 = µ2 , since the family Γ generates E. Now let α = ω0 , A = {x ∈ X : ( f1 (x), . . . , fn (x)) ∈ S}, ν j be an image of a measure µ j for a mapping x 7→ ( f1 (x), . . . , fn (x)), where S ∈ E(Kn ), f j ∈ X ֒→ X ∗ . Then νˆ 1 (y) = µˆ 1 (y1 f1 + · · · + yn fn ) = µˆ 2 (y1 f1 + · · · + yn fn ) = νˆ 2 (y) for each y = (y1 , . . . , yn ) ∈ Kn , consequently, ν1 = ν2 on E. Further compositions of f ∈ Γ with continuous functions g : K → Ks generate a family of Ks -valued functions correspondingly separating points of X (see also Chapter 9 in [Roo78]). 2.7. Proposition. Let µl and µ be measures in M(Xl ) and M(X) respectively, where Xl = c0 (αl , K), αl ≤ ω0 , X = ∏n1 Xl , n ∈ N. Then the condition µˆ (z1 , . . . , zn ) = ∏nl=1 µˆ l (zl ) for each (z1 , . . . , zn ) ∈ X ֒→ X ∗ is equivalent to µ = ∏nl=1 µl .R Proof. Let µ = ∏nl=1 µl , then µˆ (z1 , . . . , zn ) = X χe (∑ zl (xl )) ∏nl=1 µl (dxl ) = R ∏nl=1 Xl χe (zl (xl ))µl (dxl ). The reverse statement follows from Theorem 2.6. 2.8. Proposition. Let X be a Banach space over K; suppose µ, µ1 and µ2 are probability measures on X. Then the following conditions are equivalent: µ is the convolution of two measures µ j , µ = µ1 ∗ µ2 , and µˆ (z) = µˆ 1 (z)ˆµ2 (z) for each z ∈ X. Proof. Let µ = µ1 ∗ µ2 . This means by the definition that µ is the image of the measure R µ1 ⊗ µ2 for the mapping (x1 ,Rx2 ) 7→ x1 + x2 , x j ∈ X, consequently, µˆ (z) = X×X χe (z(x1 + x2 )) (µ1 ⊗µ2 )(d(x1 , x2 )) = ∏2l=1 X χe (z(xl ))µl (dxl ) = µˆ 1 (z)ˆµ2 (z). On the other hand, if µˆ 1 µˆ 2 = µ, then µˆ = (µ1 ∗ µ2 )∧ and due to the basic theorem about the Fourier-Stieltjes transform (see above) for measures with values in Ks , we have µ = µ1 ∗ µ2 . 2.9. Corollary. Let ν be a probability measure on B f (X) and µ ∗ ν = µ for each µ with values in the same field, then ν = δ0 . Proof. If z0 ∈ X ֒→ X ∗ and µˆ (z0 ) 6= 0, then from µˆ (z0 )νˆ (z0 ) = µˆ (z0 ) it follows that νˆ 0 (z0 ) = 1. From Property 2.6(5) we get that there exists m ∈ N with µˆ (z) 6= 0 for each z with kzk = p−m , since µˆ (0) = 1. Then νˆ (z + z0 ) = 1, that is, νˆ |(B(X,z0 ,p−m )) = 1. Since µ are arbitrary we get νˆ |X = 1, that is, ν = δ0 due to § 2.5. 2.10. Corollary. Let X and Y be Banach spaces over K, µ and ν be probability measures on X and Y respectively, suppose T : X → Y is a continuous linear operator. A measure ν is an image of µ for T if and only if νˆ = µˆ ◦ T ∗ , where T ∗ : Y ∗ → X ∗ is an adjoint operator. Proof. It follows immediately from § 2.5 and § 2.6. 2.11. Proposition. For a completely regular space X with the zero small inductive dimension ind(X) = 0 the following statements are accomplished: (a). if (µβ ) is a bounded net of measures in M(X) that weakly converges to a measure µ in M(X), then (ˆµβ ( f )) converges to µˆ ( f ) for each continuous f : X → K; if X is separable and metrizable then (ˆµβ ) converges to µˆ uniformly on subsets that are uniformly equicontinuous in C(X, K); (b). if M is a bounded dense family in a ball of the space M(X) for measures in M(X), then a family (ˆµ : µ ∈ M) is equicontinuous on a locally K-convex space C(X, K) in a topology of uniform convergence on compact subsets S ⊂ X. Proof. (a). Functions χe ( f (x)) are continuous and bounded on X, where µˆ ( f ) = R χ X e ( f (x))µ(dx). Then (a) follows from the definition of the weak convergence, since spanUs {χe ( f (x)) : f ∈ C(X, K} is dense in C(X, Us ). (b). For each c > 0 there exists a compact subset S ⊂ X such that kµ|(X\S) k < c/4 for

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Ks -valued measures. Therefore, for µ ∈ M and f ∈ C(X, K) with | f (x)|K < c < 1 for x ∈ S R we get |µ(X) − µˆ ( f )| = | X (1 − χe ( f (x))µ(dx)| < c/2 for Ks -valued µ, since for c < 1 and x ∈ S we have χe ( f (x)) − χe (− f (x)) = 0. 2.12. Theorem. Let X be a Banach space over K, γ : Γ → Cs be a continuous positive definite function, (µβ ) be a bounded weakly relatively compact net in the space Mt (X) of Radon norm-bounded measures and there exists limβ µˆ β ( f ) = γ( f ) for each f ∈ Γ and ˜ where Γ ⊂ C(X, K) is a vector subspace uniformly on compact subsets of the completion Γ, separating points in X. Then (µβ ) weakly converges to µ ∈ Mt (X) with µˆ |Γ = γ. Proof. Is analogous to the proof of Theorem I.2.17 and follows from Theorem 2.6 and using the non-Archimedean Lebesgue convergence theorem recalled above. 2.13. Theorem. (a). A bounded family of measures in M(Kn ) is weakly relatively compact if and only if a family (ˆµ : µ ∈ M) is equicontinuous on Kn . (b). If (µ j : j ∈ N) is a bounded sequence of measures in Mt (Kn ), γ : Kn → Us is a continuous function, µˆ j (y) → γ(y) for each y ∈ Kn uniformly on compact subsets in Kn , then (µ j ) weakly converges to a measure µ with µˆ = γ. (c). A bounded sequence of measures (µ j ) in Mt (Kn ) weakly convereges to a measure µ in Mt (Kn ) if and only if for each y ∈ Kn there exists lim j→∞ µˆ j (y) = µˆ (y). (d). If a bounded net (µβ ) in Mt (Kn ) converges uniformly on each bounded subset in Kn , then (µβ ) converges weakly to a measure µ in Mt (Kn ), where n ∈ N. Proof. (a). This follows from Proposition 2.11. (b). Due to the non-Archimedean Fourier transform and the Lebesgue convergence theorem for Ks -valued measures formulated above and from the condition limR→∞ sup|y|>R |γ(y)|Rn = 0 it follows, that for each ε > 0 there exists R0 > 0 such that limm sup j>m kµ j |{x∈Kn :|x|>R} k ≤ 2 sup|y|>R |γ(y)|R < ε for each R > R0 . In view of Theorem 2.12 (µ j ) converges weakly to µ with µˆ = γ. (c,d). These can be proved analogously to I.2.18. 2.14. Corollary. If (ˆµβ ) → 1 uniformly on some neighborhood of 0 in Kn for a bounded net of measures µβ in Mt (Kn ), then (µβ ) converges weakly to δ0 . 2.15. Definition. A family of probability measures M ⊂ Mt (X) for a Banach space X over K is called planely concentrated if for each c > 0 there exists a K-linear subspace S ⊂ X with dimK S = n < ℵ0 such that inf(kSc kµ : µ ∈ M) > 1 − c. The Banach space Mt (X) is supplied with the following norm kµk 2.16. Theorem. Let X be a Banach space over K with a family Γ ⊂ X ∗ separating points in M ⊂ Mt (X). Then M is weakly relatively compact if and only if a family {µz : µ ∈ M} is weakly relatively compact for each z ∈ Γ and M is planely concentrated, where µz is an image measure on K of a measure µ induced by z. Proof. It follows from the Alaoglu-Bourbaki theorem recalled above and Lemmas I.2.5 and I.2.21. 2.17. Theorem. For X and Γ the same as in Theorem 2.16 a sequence {µ j : j ∈ N} ⊂ Mt (X) is weakly convergent to µ ∈ Mt (X) if and only if for each z ∈ Γ there exists lim j→∞ µˆ j (z) = µˆ (z) and a family {µ j } is planely concentrated. Proof. It follows immediately from Theorems 2.12,13,16. 2.18. Proposition. Let X be a weakly regular space with ind(X) = 0, Γ ⊂ C(X, K) be a vector subspace separating points in X, (µn : n ∈ N) ⊂ Mt (X), µ ∈ Mt (X), limn→∞ µˆ n ( f ) = µˆ ( f ) for each f ∈ Γ. Then (µn ) is weakly convergent to µ relative to the weakest topology

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σ(X, Γ) in X relative to which all f ∈ Γ are continuous. Proof. It follows from Theorem 2.13. 2.19. Let (X, U) = ∏λ (Xλ , Uλ ) be a product of measurable completely regular Radon spaces (Xλ , Uλ ) = (Xλ , Uλ , Kλ ), where Kλ are compact classes approximating from below each measure µλ on (Xλ , Uλ ), that is, for each c > 0 and elements A of an algebra Uλ there is S ∈ Kλ , S ⊂ A with kA \ Skµλ < c. Theorem. Each bounded quasi-measure µ with values in Ks on (X, U) (that is, µ|Uλ is a bounded measure for each λ) is extendible to a measure on an algebra A f (X, µ) ⊃ U, where an algebra U is generated by a family (Uλ : λ ∈ Λ). Proof. We have 2.1(i) by the condition and kXkµ < ∞, if 2.1(iii) is satisfied. It remains T to prove 2.1(iii). For each sequence (An ) ⊂ U with n An = 0/ and each c > 0 for each j ∈ N we choose K j ∈ K, where the compact class K is generated by (Kλ ) (see also Proposition T T / then there 1.1.8[DF91]), such that K j ⊂ A j and kA j \ K j kµ < c. Since ∞ K ⊂ n An = 0, n=1 Tl Tl Sl n / hence Al = Al \ n=1 Kn ⊂ n=1 (An \ Kn ), consequently, exists l ∈ N with n=1 Kn = 0, kAl kµ ≤ maxn=1,...,l (kAn \ Kn kµ ) < c. Remind a theorem about uniqueness of extensions of measures (see also Theorem 7.8[Roo78]). Let µ be a measure on R . Let S be a separating covering ring of X such that S is a sub-ring in R µ and let ν be a restriction of µ on S . Then S ν = R µ and ν¯ = µ¯ . To finish the proof of this theorem it remains to use the theorem about uniqueness of an extension of a measure. 2.19.1. Note. More general theorem is given in § 2.37, since products are particular cases of projective limits (see also § § 2.36 and 2.38). 2.20. Definition. Let X be a Banach space over K, then a mapping f : X → Us is called pseudo-continuous, if its restriction f |L is uniformly continuous for each K-linear subspace L ⊂ X with dimK L < ℵ0 . Let Γ be a family of mappings f : Y → K of a set Y into a field K. We denote by C(Y, Γ) an algebra of subsets of the form C f1 ,..., fn ;E := {x ∈ X : ( f1 (x), . . . , fn (x)) ∈ S}, where S ∈ Bco(Kn ), f j ∈ Γ. We supply Y with a topology τ(Y ) which is generated by a base (C f1 ,..., fn ;E : f j ∈ Γ, E is open in Kn ). 2.21. Theorem. Non-Archimedean analog of the Bochner-Kolmogorov theorem. Let X be a Banach space over K, X a be its algebraically dual K-linear space (that is, of all linear mappings f : X → K not necessarily continuous). A mapping θ : X a → Us is a characteristic functional of a probability measure µ with values in Ks and it is defined on C(X a , X) if and only if θ satisfies Conditions 2.5(3, 5) for (X a , τ(X a ) and is pseudocontinuous on X a , where Ks ∪ Cs ⊂ Us . Proof. (I). For dimK X = card(α) < ℵ0 a space X a is isomorphic with Kα , hence the statement of this theorem for a measure µ with values in Ks follows from the basic theorem about the Fourier-Stieltjes transform reminded above and Theorems 2.6 and 2.13, since θ(0) = 1 and |θ(z)| ≤ 1 for each z. (II). Now let α = ω0 . It remains to show that the conditions imposed on θ are sufficient, because their necessity follows from the modification of § 2.5 (since X has an algebraic embedding into X a ). The space X a is isomorphic with KΛ which is the space of all K-valued functions defined on the Hamel basis Λ in X. Let J be a family of all non-void subsets in Λ. For each A ∈ J there exists a functional θA : KA → C such that θA (t) = θ(∑y∈A t(y)y) for t ∈ KA . From the conditions imposed on θ it follows that θA (0) = 1, θA is uniformly continuous and bounded on KA , moreover, due to 2.5(5) for each c > 0 there are n and q > 0

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such that for each j > n and z ∈ KA the following inequality is satisfied: (i) |θA (z) − θ j (z)| ≤ cbq, moreover, L j ⊃ KA , q is independent from j, c and b. From (I) it follows that on B f (KA ) there exists a probability measure µA such that µˆ A = θA . The family of measures {µA : A ∈ J} is consistent and bounded, since µA = µE ◦ (PEA )−1 , if A ⊂ E, where PEA : KE → KA are the natural projectors. Indeed, this is accomplished due to Conditions (i), 2.5(5) for X a and due to the basic theorem about the Fourier-Stieltjes transform. In view of Theorem 2.19 on a cylindrical algebra of the space KΛ there exists the unique measure µ such that µA = µ ◦ (PA )−1 for each A ∈ J, where PA : KΛ → KA are the natural projectors. From X a = KΛ it follows that µ is defined on C(X a , X). For µ on C(X a , X) there exists its extension on A f (X, µ) such that A f (X, µ) ⊃ Bco(X) (see § 2.1). 2.22. For f ∈ L(X, µ, Us ) and Ks -valued measure µ let Z

X

f (x)µ∗ (dx) = lim

Z

n→∞ X

gn (x)µ∗ (dx)

for norm-bounded sequence of cylindrical functions gn from L(X, µ, Us ) converging to f uniformly on compact subsets of X, where Ks ⊂ Us . Due to the Lebesgue converging theorem this limit exists and does not depend on a choice of {gn : n}. Lemma. A sequence of weak distributions (µLn ) of probability Radon measures is generated by a Ks -valued probability measure µ on Bco(X) of a Banach space X over K if and only if there exists Z (i) lim

|ξ|→∞ X

where

R

X

Gξ (x)µ∗ (dx) = 1,

Gξ (x)µ∗ (dx) := Sξ ({µLn : n}) and Sξ ({µLn }) := lim

Z

n→∞ Ln

Fn (γξ,n )(x) µLn (dx),

γξ,n (y) :=

m(n)

∏ γξ (yl ), l=1

Fn is a Fourier transformation by (y1 , . . . , yn ), y = (y j : j ∈ N), y j ∈ K, γξ (y) := C(ξ)s−2 min(0,ord p (y,ξ)) , C(ξ) ∈ Ks , γξ : K → Ks , y, ξ ∈ K, νξ (K) = 1,νξ (dy) = γξ (dy)w(dy), S w : Bco(K) → Ks is the Haar measure; here m(n) = dimK Ln < ℵ0 , cl( n Ln ) = X = c0 (ω0 , K). Proof is quite analogous to that of § I.2.30 with the substitution of Z Gξ (x)µ∗ (dx) − 1 < c/2 X

for real-valued measures on |kGξ (x)k − 1| < c/2 for Ks -valued measures. 2.23. Notes and definitions. Suppose X is a locally convex space over a locally compact field K with non-trivial non-Archimedean normalization and X ∗ is a topologically dual space. For a Ks -valued measure µ on X a completion of a linear space of characteristic functions {chU : U ∈ Bco(X)} in L(X, µ, Us ) is denoted by Bµ (X), where Ks ⊂ Us . Then X is called a KS-space if on X ∗ there exists a topology τ such that the continuity of f : X ∗ → Us with k f kC0 < ∞ is necessary and sufficient for f to be a characteristic functional of a tight

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measure of the finite norm, where Cs ⊂ Us . Such topology is called the K-Sazonov type topology. The class of KS-spaces contains all separable locally convex spaces over K. For example, l ∞ (α, K) = c0 (α, K)∗ . In particular we also write c0 (K) := c0 (ω0 , K) and l ∞ (K) := l ∞ (ω0 , K), where ω0 is the first countable ordinal. Let nK (l ∞ , c0 ) denotes the weakest topology on l ∞ for which all functionals px (y) := supn |xn yn | are continuous, where x = ∑n xn en ∈ c0 and y = ∑n yn e∗n ∈ l ∞ , en is the standard base in c0 . Such topology nK (l ∞ , c0 ) is called the normal topology. The induced topology on c0 is denoted by nK (c0 , c0 ). 2.23.1. Proposition. (i). The topology nK (l ∞ , c0 ) is the K-locally convex Hausdorff topology on l ∞ (K) and the family of sets U := {Ux : x ∈ c0 }, where Ux := {z ∈ l ∞ : px (z) < 1}, is the base of neighborhoods of zero for it. (ii). The topology nK (l ∞ , c0 ) is strictly weaker than the norm topology in l ∞ . (iii). The space c0 is dense in l ∞ in the topology nK (l ∞ , c0 ). Proof. (i). If G is a T0 -group, then G is a Hausdorff group (see Chapter 1 in [HR79]). Since l ∞ has the additive group structure and nK (l ∞ , c0 ) is the T0 -topology, hence l ∞ is Hausdorff in this topology. By the definition U forms the base of neighborhoods of zero. (ii). To prove it consider the standard orthonormal base {en : n} in c0 , then limn→∞ px (en ) = |xn |K = 0 for each x = (xn : n) ∈ c0 , where xn ∈ K for each n ∈ N. (iii). Put an (x) := (x1 , . . . , xn , 0, 0, . . . ) for each x ∈ l ∞ . Then limn→∞ pz (an (x) − x) = limn→∞ supn∈N |xn+1 yn+1 |K = 0 for each y ∈ c0 . 2.24. Theorem. Let f : l ∞ (K) → Us be a functional such that (i) f (0) = 1 and k f kC0 ≤ 1, (ii) f is continuous in the normal topology nK (l ∞ , c0 ), then f is the characteristic functional of a probability measure on c0 (K). Proof. If ν is the Haar measure on Kn , then on Bco(Kn ) it takes values in Q, when it is chosen with ν(B(Kn , 0, 1)) = 1 and hence ν(B(Kn , x, pn )) = pn for each x ∈ Kn and each n ∈ Z. If K is a locally compact non-Archimedean field, R is its maximal compact sub-ring B(K, 0, 1) and P is the maximal ideal of R, then the order of a nontrivial character χ of K is the largest integer n ∈ Z such that χ|P−n = 1, it is denoted by ord(χ) := n. In view of Proposition 12 [Wei73] if χ is a non-trivial character of K of order n, then for each m ∈ Z, χ(xt) = 1 for each t ∈ Pm if and only if x ∈ P−n−m . Therefore, for a characteristic function ChB of a ball B we have ChB(Kn ,0,|π|m ) (x1 , . . . , xn ) = [ν(B(Kn , 0, |π|−m ))]−1 ! Z n

Kn

ChB(Kn ,0,|π|−m ) (y1 , . . . , yn )χ

∑ xl yl

ν(dy),

l=1

since B(Kn , 0, |π|−m ) = (B(K, 0, |π|−m ))n , where P = πR, |π| is the generator of the normalization group ΓK . Therefore, m

m

P{|V1 |K < |π| , . . . , |Vn |K < |π| } = =

Z

Ω

(

[ν(B(Kn , 0, |π|−m ))]−1

Z

Kn

Z

Kn

ChB(Kn ,0,|π|m ) (V1 (ω), . . . ,Vn (ω))P(dω)

)  n  ChB(Kn ,0,|π|−m ) (y)χ ∑ yl Vl (ω) ν(dy) P(dω), l=1

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hence (i) P{|V1 |K < |π|m , . . . , |Vn |K < |π|m } = [ν(B(Kn , 0, |π|−m ))]−1

Z

Kn

fW (y)ChB(Kn ,0,|π|−m ) (y)ν(dy)

for measurable maps V j : (Ω, B, P) → (K, Bco(K)), where (Ω, B, P) is a probability space for a probability measure P with values in Ks on an algebra B of subsets of a set Ω, fW is R a characteristic function of W = (V1 , . . . ,Vn ), fW (y) := Ω χW (ω) (y)P(dω). To continue the proof we need the following statements. 2.25. Lemma Let f : c0 (K) → Cs be a function satisfying the following two conditions: (i) | f (x)| ≤ 1 for each x ∈ c0 (K), (ii) f is continuous at zero in the topology nK (c0 , c0 ), then for each ε > 0 there exists λ(ε) ∈ c0 (K) such that |1 − f (x)| < pλ(ε) (x) + ε for each x ∈ c0 (K). Proof. In view of continuity for each ε > 0 there exists y(ε) ∈ c0 such that |1 − f (x)| < ε −1 if py(ε) < 1. Put λ(ε) = π−1 K y(ε), where πK ∈ K is such that |πK | = p . If x ∈ c0 is such that pλ(ε) (x) < p−1 , then |1 − f (x)| < ε ≤ ε + pλ(ε) (x). If pλ(ε) (x) ≥ p, then |1 − f (x)| ≤ 2 ≤ p < pλ(ε) (x) + ε. 2.26. Lemma. Let {Vn : n ∈ N} be a sequence of K-valued random variables for P with values in Ks . If for each β > 0 and ε > 0 there exists Nε ∈ N such that (i) kP|{supn≥Nε |Vn |K ≤β} k ≥ 1 − ε(1 + β−1 ), then limn Vn = 0 P-almost everywhere on Ω. Proof. Consider a marked β > 0 and for each ε > 0 there exists Nε ∈ N such that Inequality (i) is satisfied. Take a sequence {εn : n ∈ N} such that 0 < εn+1 < εn for each n and limn→∞ εn = 0, then there exists a sequence {kn := Nεn : n} such that kn+1 ≥ kn for each n ∈ N. Put an (ω) := supm≥kn |Vn (ω)|K . Then an ≥ 0 and an+1 ≤ an for each n ∈ N. Hence there exists limn→∞ an (ω) = infn supm≥kn |Vn (ω)|K =: X(ω). Since ΓK is discrete in (0, ∞) for the local field K, then X(ω) is the discrete real random variable on (Ω, B, P). Consider bn (ω) := supm≥n |Vm (ω)|K , consequently, bn ≥ 0 and bn+1 ≤ bn for each n ∈ N. Thus there exists limn→∞ bn and {an : n} is the subsequence of {bn : n}, hence limn→∞ bn = limn→∞ an = X. From the definition of an we have kP|an ≤β k ≥ 1 − εn (1 + β−1 ). Since P is the probability measure, then kP|(limn→∞ an )≤β k = 1 and kP|(limn→∞ bn )≤β k = 1 for each β > 0. Therefore, kP|(limn→∞ bn =0) k = 1 and inevitably limn→∞ bn = 0 P-almost everywhere. 2.27. Proposition. Let f : c0 (K) → Us be a function such that (i) f (0) = 1 and | f (x)| ≤ 1 for each x ∈ c0 , (ii) f (x) is continuous in the normal topology nK (c0 , c0 ). Then there exists a probability measure µ on c0 (K) such that f (x) = µˆ (x) for each x ∈ c0 . Proof. Consider functions fn (x1 , . . . , xn ) := f (x1 e1 +· · ·+xn en ), where x = ∑ j x j e j ∈ c0 . From Condition (ii) and Proposition 2.23.1.(ii) it follows, that f (x) is continuous in the norm topology. The field K is spherically complete and X has a basis orthonormal in the nonArchimedean sense. So there are canonical embeddings of Kn into X and the corresponding projections from X onto such finite dimensional over K subspaces associated with the basis.

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From Chapters 7,9 [Roo78] it follows, that there exists a consistent family of tight measures µn on Kn such that µˆ n (x) = fn (x) for each x ∈ Kn . In view of Theorem 2.19 there exists a probability space (Ω, B, P) with a Ks -valued measure P and a sequence of random variables {Vn } such that µn (A) = P{ω ∈ Ω : (V1 (ω), . . . ,Vn (ω)) ∈ A} for each clopen subset A in Kn , consequently, limn Vn = 0 P-almost everywhere in Ω. In view of the preceding lemmas we have the following inequality: |1 − kP|(|Vn | 0 and δ > 0 there are a finite-dimensional over K subspace L in X and a compact subset W ⊂ X such that W ⊂ Lδ , kµ|(X\W ) k < b, hence kµ|(X\Lδ ) k < b. We consider the expression J( j, l) (see § I.2.35). and the compact operator S : X → X with e˜ j (Sel ) = ξ j,l t. Then |θ(0) − θ(z)| < c/2 + |˜z(Sz)| < c for the Ks -valued measure, if |˜z(Sz)| < |t|c/2, where Qs ⊂ Ks ⊂ Us . We choose r such that kµ|(X\B(X,0,r)) k < c/2 with S corresponding to (r j : j), where r1 = r, L1 = L, then we take t ∈ K with |t|c = 2. (I → II). Without restriction of generality we may take θ(0) = 1 after renormalization of non-trivial θ. In view of Theorem 2.24 as in § 2.5 we construct using θ(z) a consistent family of finite-dimensional distributions {µLn : n} all with values in Ks . Let mLn be the Ks -valued Haar measure on Ln which is considered as Qap with a = dimK Ln dimQp K, m(B(Ln , 0, 1)) = 1. If Sc is a compact operator such that |θ(y) − θ(x)| < c for |˜z(Sc z)| < 1, z = x − y, then |1 − θ(x)| < max(C, 2|x(S ˜ c x)|) and kγξ,n (z)(1 − θ(z))kmLn ≤ max(kγξ,n (z)kmLn C, 2|(γξ,n (z))˜z(Sc z)|mLn ) ≤ max(C, bkSc k/|ξ|2 ), where b := p × sup|ξ|>r (|ξ|2 kγξ,n (z)z2 kmLn ) < ∞ for the Ks -valued measures. Remind the substitution theorem for integrals (see also [Kob77, Roo78, Sch84, Wei73]). Let U an V be compact open subsets in K and let σ : U → V be a C1 -homeomorphism, σ′ (x) 6= 0 for all x ∈R U. Let f : V → K be a continuous function, λ : Bco(K) → Ks be a R s Haar measure, then V f (y)λ(dy) = U f (σ(x))modK (σ′ (x))λ(dx). Due to the formula of changing variables in integrals the following equality is valid: |1 − kGξ (x)kµ∗ | ≤ max(C, bkSc k/|ξ|2 ) for the Ks -valued measures. Then taking the limit with |ξ| → ∞ and then with c → +0 with the help of Lemma 2.22 we get the statement (I → II). 2.31. Definition. Let on a completely regular space X with ind(X) = 0 two non-zero Ks -valued measures µ and ν are given. Then ν is called absolutely continuous relative to µ R if there exists f such that ν(A) = A f (x)µ(dx) for each A ∈ Bco(X), where f ∈ L(X, µ, Ks ) and it is denoted ν ≪ µ. Measures ν and µ are singular to each other if there is F ∈ E with kX \ Fkµ = 0 and kFkν = 0 and it is denoted ν ⊥ µ. If ν ≪ µ and µ ≪ ν then they are called equivalent, ν ∼ µ. 2.32. Definition and note. For µ : E(X) → Ks a sequence (φn (x) : n) ⊂ L(µ) is called a martingale if for each ψ ∈ L(µ|Un ): (i)

Z

X

φn+1 (x)ψ(x)µ(dx) =

Z

X

φn (x)ψ(x)µ(dx)

such that (φn : n) is uniformly converging on A f (X, µ)-compact subsets in X, where Un is the minimal algebra such that (φ j : j = 1, . . . , n) ⊂ L(µ|Un ), µ|Un is a restriction of µ on Un ⊂ E(X), X is the Banach space over K. Recall Lemma 7.10 [Roo78]. Let µ be a measure on R . For b > 0 put Xb := {x : Nµ (x) ≥ b}. Then the restrictions of the R - and the R µ -topologies on Xb coincide. A function f : X → K is R µ -continuous if and only if for every b > 0 the restriction of f to Xb is R -continuous.

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Remind as well Theorem 7.12 [Roo78]. Let µ be a measure on R . A function f : X → K is µ-integrable if and only if it satisfies the following two properties. (Ri). f is R µ -continuous. (Rii). For each b > 0 the set {x : | f (x)|Nµ (x) ≥ b} is R µ -compact, hence contained in some {x : Nµ (x) ≥ δ} with δ > 0. In view of these two statements formulated just above for kXkµ < ∞ the A f (X, µ)topology on compact subspaces Xc := [x ∈ X : Nµ (x) ≥ c] coincides with the initial topology, if µ is defined on E such that Bco(X) ⊂ E ⊂ A f (X, µ), where c > 0. 2.33. Theorem. If there is a martingale (φn : n) for µ with values in Ks and supn kφn kNµ < ∞, then there exists limn→∞ φn (x) =: φ(x) ∈ L(µ). Proof. Let ψ(x) be a characteristic function of a clopen subset in X, then for each φn j j there exists a sequence of simple functions (φn : j ∈ N ) such that lim j→∞ kφn − φn kNµ = 0. j(n)

j(n+1) j(n) X (φn+1 (x) − φn )ψ(x)µ(dx)| < j(n+1) j(n) kφn+1 − φn kNmu < c and there exists the Lebesgue theorem, if (c = c(n) = s−n :

From kφn − φn kNµ < c and 2.32.(i) it follows that | ckψkNµ for each ψ ∈ L(µ), consequently, j(n)

R

limn→∞ φn = limn→∞ φn = φ ∈ L(µ) due to j(n) n ∈ N), where for each φn is chosen j(n) ∈ N, since (φn : n) is a Cauchy sequence in the Banach space L(µ) due to the ultra-metric inequality. 2.34. Proposition. Let (X, R , µ) be a measure space. Then there exists a quotient mapping π : X → Y on a Hausdorff zero-dimensional space (Y, τG ) and π(µ) := ν is a measure on Y such that G = π(R ), where (Y, G , ν) is a measure space. Proof. Suppose that (Y, τG ) is a T0 -space, where G is a covering ring of Y . For each two distinct points x and y in Y there exists a clopen subset U in Y such that either x ∈ U and y ∈ Y \U or y ∈ U and x ∈ Y \U, since the base of topology τG in Y consists of clopen subsets. On the other hand, Y \ U is also clopen, since U is clopen. Therefore, Y is the Hausdorff space. Clearly this implies that Y is the Tychonoff space (see also § 6.2 [Eng86], but it is necessary to note that we consider the definition of the zero-dimensional space more general without T1 -condition in § 2.1.2). Now we construct a T1 -space Y , that is a quotient space of X. For this consider the relation in X: xκy if and only if for each S ∈ R with x ∈ S there is the inclusion {x, y} ⊂ S. Evidently, T xκx, that is, κ is reflexive. The relation xκy means, that y ∈ Vx := x∈S∈R S, where Vx is closed in X, then from y ∈ S it follows, that x ∈ S, since otherwise y ∈ / Vx , because R is a covering ring. Therefore, Vx = Vy and yκx, hence κ is symmetric. Let xκy and yκz, then Vx = Vy = Vz , consequently, xκz and κ is transitive. Therefore, κ is the equivalence relation. Let π : X → Y := X/κ be the quotient mapping and Y be supplied with the zerodimensional topology generated by the covering ring G such that π−1 (G ) = R , since each A ∈ R is clopen, then for each x ∈ A ∈ R we have Vx ⊂ A. Then π−1 ([y]) = Vy for each y ∈ X and [y] := π(y). Hence each point [y] ∈ Y is closed, hence Y is the T1 -space. The topology in Y is generated by the covering ring G , consequently, Y is the Hausdorff space (see above), since from the T1 separation property it follows the T0 separation property. If S is a shrinking family with the void intersection in Y such that S ⊂ G , then π−1 (S ) is also a shrinking family with the void intersection in X such that π−1 (S ) ⊂ R , hence from limA∈π−1 (S ) µ(A) = 0 it follows limA∈S ν(A) = 0. Therefore, Condition (iii) from § 2.1 is satisfied. Evidently, kνk = kµk and ν is additive on G , hence ν is the measure.

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2.35. Note. In view of Proposition 2.34 we consider henceforth Hausdorff zerodimensional measurable (X, R ) spaces if another is not specified. In the classical case the principal role in stochastic analysis plays the Kolmogorov theorem, that gives the possibility to construct a stochastic process on the basis of a system of finite dimensional real-valued probability distributions (see § III.4 [Kol56]). The following theorems resolve this problem for Ks -valued measures in cases of a product of measure spaces, a consistent family of measure spaces and in cases of bounded cylindrical distributions. Consider now a family of probability measure spaces {(X j , R j , µ j ) : j ∈ Λ}, where Λ is a set. Suppose that each covering ring R j is complete relative to a measure µ j , that is, R j = R µ j , where R µ j denotes a completion of R j relative to µ j . Let X := ∏ j∈Λ X j be the product of topological spaces supplied with the product, that is Tychonoff, topology τX , where each X j is considered in its τR j -topology. There is the natural continuous projection S T π j : X → X j for each j ∈ Λ. Let R be the ring of the form j1 ,..., jn ∈Λ,n∈N nl=1 π−1 jl (R jl ). 2.36. Note. A set Λ is called directed if there exists a relation ≤ on it satisfying the following conditions: (D1) If j ≤ k and k ≤ m, then j ≤ m; (D2) For every j ∈ Λ, j ≤ j; (D3) For each j and k in Λ there exists m ∈ Λ such that j ≤ m and k ≤ m. A subset ϒ of Λ directed by ≤ is called co-final in Λ if for each j ∈ Λ there exists m ∈ ϒ such that j ≤ m. Suppose that Λ is a directed set and {(X j , R j , µ j ) : j ∈ Λ} is a family of probability measure spaces, where R j is the covering ring (not necessarily separating). Supply each X j with a topology τ j such that its base is a ring R j . Let this family be consistent in the following sense: (1) there exists a mapping πkj : Xk → X j for each k ≥ j in Λ such that (πkj )−1 (R j ) ⊂ R k , k m π j (x) = x for each x ∈ X j and each j ∈ Λ, πm k ◦ πl = πl for each m ≥ k ≥ l in Λ; k (2) πl (µk ) = (µl ) for each k ≥ l in Λ. Such family of measure spaces is called consistent. 2.37. Theorem. Let {(X j , R j , µ j ) : j ∈ Λ} be a consistent family as in § 2.36. Then there exists a probability measure space (X, R µ , µ) and a mapping π j : X → X j for each j ∈ Λ such that (π j )−1 (R j ) ⊂ R and π j (µ) = µ j for each j ∈ Λ. Proof. We have (πkj )−1 (R j ) ⊂ R k for each k ≥ j in Λ, then (πkj )−1 (τ j ) ⊂ τk for each k ≥ j in Λ, since each open subset in (X j , τ j ) is a union of some subfamily G in R j and S S (πkj )−1 ( G ) = A∈G (πkj )−1 (A). Therefore, each πkj is continuous and there exists the inverse system S := {Xk , πkj , Λ} of the spaces Xk . Its limit lim S =: X is the topological space with a topology τX and continuous mappings π j : X → X j such that πkj ◦ πk = π j for each k ≥ j in Λ (see also § 2.5 in [Eng86]). Each element x ∈ X is a thread x = {x j : x j ∈ X j for each j ∈ Λ, πkj (xk ) = x j for each k ≥ j ∈ Λ}. Then π−1 j (R j ) =: G j is the ring of subsets in X such that G j ⊂ τX for each j ∈ Λ. The base of topology of (X, τX ) is formed by subsets π−1 j (A), where A ∈ τ j , j ∈ Λ, but R j is the base of topology τ j for each j, hence S {B : B = π−1 j∈Λ G j is the j (A), A ∈ R j , j ∈ Λ} is the base of τX . Therefore, the ring R := base of τX . In view of Proposition 2.34 we can reduce our consideration to the case, when each R j is separating on X j and R is separating on X, since G j ⊂ G k for each k ≥ j in Λ. Consider on R a function µ with values in K such that µ(π−1 j (A)) := µ j (A) for each A ∈ R j and each j ∈ Λ. If A and B are disjoint elements in R , then there exists j ∈ Λ such j

Non-Archimedean Valued Measures

107

that A and B are in G j , hence −1 (i) A = π−1 j (C) and B = π j (D) for some C and D in R j , consequently, µ(A ∪ B) = µ j (C ∪ D) = µ j (C) + µ j (D) = µ j (A) + µ j (B), that is, µ is additive. Moreover, kAkµ = kCkµ j for each A = π−1 j (C) with C ∈ R j , hence kXkµ = 1. Since µ(X) = µ j (X j ) and µ j (X j ) = 1 for each j ∈ Λ, then µ(X) = 1. Therefore, µ satisfies Conditions 2.1.(i, ii). It remains to verify Condition 2.1.(iii). By formula of § 2.1 we have the function Nµ (x) on (X, R ) such that for each x ∈ X and ε > 0 there exists A ∈ R such that (ii) kAkµ − ε < Nµ (x) ≤ kAkµ . In view of (i) and upper semi-continuity of Nµ j (x j ) on (X j , R j ) for each x ∈ X and ε > 0 there exists j ∈ Λ and its neighborhood A = π−1 j (C) ∈ R such that (iii) Nµ j (y j ) < Nµ (x) + ε for each y ∈ A, where y j := π j (y). Hence for each x ∈ X and each ε > 0 there exists a basic neighborhood A of x such that (iv) Nµ (y) < Nµ (x) + ε for each y ∈ A, that is, Nµ (x) is upper semi-continuous on (X, R ), since 0 ≤ Nµ j (x j ) ≤ 1 for each x j ∈ X j and j ∈ Λ. From Formulas (i, ii, iii) and 2.1.(ii) we have (v) kAkµ = supx∈X Nµ (x) for each A ∈ R , since kAkµ = supx∈C Nµ j (x). For a compact subset V in X and each ε > 0 there exists a finite covering {E1 , . . . , Em } ⊂ R of V such that inequalities (ii − iv) are satisfied for each El instead of A. Therefore, (vi) supx∈V Nµ (x) ≤ maxl=1,...,m kEl kµ ≤ supx∈V Nµ (x) + 2ε and (vii) supx∈V Nµ (x) = infR ∋A⊃V kAkµ . Though the compact subset V is not necessarily in R we take Equation (vii) as the definition of kV kµ := infR ∋A⊃V kAkµ . Choose a sequence ε j = δ > 0 for each j ∈ Λ, where δ > 0 is independent from j. For each ε j > 0 a subset X j,ε j := {x j : x j ∈ X j , Nµ j (x j ) ≥ ε j > 0} is compact. If xk ∈ Xk,εk , then Nµ j (πkj (xk )) ≥ εk for each j < k, since (πkj )−1 (R j ) ⊂ R k and kBkµk ≤ kAkµk for each B and A k m in R k with B ⊂ A. Hence πkj (Xk,εk ) ⊂ X j,εk for each j ≤ k in Λ. Since πm k ◦ πl = πl for each m ≥ k ≥ l in Λ, then {Xk,δ , πkj , Λ} is the inverse mapping system. The image πkj (Xk,δ ) of each compact set Xk,δ is compact for each k > j (see also Theorem 3.1.10 [Eng86]), since each (Xk , τk ) is the Hausdorff space in our consideration. Since the limit of an inverse mapping system of compact spaces is compact (see also Theorem 3.2.13 [Eng86]), then the limit X{ε j : j} := lim{Xk,εk , πkj , Λ} is the compact subset in X such that X{ε j : j} is homeomorphic with θ(X) ∩ ∏k∈Λ Xk,εk , where θ : X ֒→ ∏k∈Λ Xk is the embedding. For a shrinking family S in R consider all finite intersections of finite families in S , this gives a centered family S 0 in R such that S ⊂ S 0 and denote it also by S . Applying (i − vii) to V = X{ε j : j} and using basic neighborhoods U = π−1 k (Uk ), where Uk ∈ R k , we get that for T / each shrinking family S in R with S = 0 there exists limA∈S kAkµ = 0, since due to (vi) we have (viii) Nµ (x) ≤ δ for each x ∈ X \ X{ε j : j} , since Nµ j (x j ) < δ for each x j ∈ X \ X j,δ and each j ∈ Λ. Using the completion of R relative to µ we get the probability measure space (X, R µ , µ). 2.38. Note. Theorem 2.37 has an evident generalization to the following case: kX j kµ j < ∞ for each j and there exist two limits lim j∈Λ0 µ j (X j ) ∈ K and lim j∈Λ kX j kµ j < ∞, where Λ0 := { j : j ∈ Λ, µ j (X j ) 6= 0} and Λ \ Λ0 is bounded in Λ. We have kX j kµ j ≤ kXk kµk for each j ≤ k in Λ, since πkj (µk ) = µ j and (πkj )−1 (R k ) ⊂ R j . Since Λ is directed, then lim j∈Λ kX j kµ j = sup j∈Λ kX j kµ j . Since a cylindrical distribution µ is defined on R

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and bounded on it, then µ has an extension to the bounded measure µ on R µ such that µ(X) = lim j∈Λ µ j (X j ) and kXkµ = lim j∈Λ kX j kµ j , where R µ is the completion of R relative to µ. Let now X be a set with a covering ring R such that X ∈ R . Let also {(X, G j , µ j ) : j ∈ Λ} be a family of measure spaces such that Λ is directed and G j ⊂ G k for each j ≤ k ∈ Λ, S R = j∈Λ G j . Suppose µ : R → K is such that µ|G j = µ j and µk |G j = µ j for each j ≤ k in Λ. Then the triple (X, R , µ) is called the cylindrical distribution. For each A ∈ R there exists j ∈ Λ such that A ∈ G j , hence kAkµ j = kAkµk for each k ≥ j in Λ, consequently, kAkµ := limk∈Λ kAkµk is correctly defined. Suppose µ is bounded, that is,kXkµ < ∞. (A particular simpler case is given below in § 2.41). 2.39. Theorem. Let (X, R , µ) be a bounded cylindrical distribution as in § 2.38. Then µ has an extension to a bounded measure µ on a completion R µ of R relative to µ. Proof. Let τX be a topology on X generated by the base R . In view of Proposition 2.34 each covering ring G j of X produces an equivalence relation κ j and a quotient mapping π j : X → X j such that π j (G j ) =: R j is a separating covering ring of X j , where X j is zerodimensional and Hausdorff. Moreover, R j is a base of a topology τ j on X j . Since G k ⊃ G j for each k ≥ j, then on (Xk , (πkj )−1 (R j )) there exists an equivalence relation κkj and a k m quotient (continuous) mapping πkj : Xk → X j such that πm k ◦ π j = π j for each j ≤ k ≤ m in Λ. Hence there exists an inverse mapping system {Xk , πkj , Λ}. Therefore, the set X in the topology τX generated by its base R consisting of clopen subsets is homeomorphic with lim{Xk , πkj , Λ}. Each π j (µ) = µ j is a bounded measure on (X j , R j ) such that πkj (µk ) = µ j and (πkj )−1 (R j ) ⊂ R k for each k ≥ j ∈ Λ. Therefore, {(X j , R j , µ j ) : j ∈ Λ} is the consistent family of measure spaces. From the definition of µ it follows that µ is additive, hence kXkµ is correctly defined. From kXkµ < ∞ it follows kX j kµ j < ∞ for each j ∈ Λ and there exists lim j∈Λ kX j kµ j = kXkµ . From X ∈ R it follows, that µ(X) = µ j (X j ) for each j ∈ Λ. Then this Theorem follows from Theorem 2.37. 2.40. Note. Let X := ∏t∈T Xt be a product of sets Xt and on X a covering ring R be given such that for each n ∈ N and pairwise distinct points t1 , . . . ,tn in a set T there exists a K-valued measure Pt1 ,...,tn on a covering ring R t1 ,...,tn of Xt1 × · · · × Xtn such t ,...,t that πt11 ,...,tn+1 (R t1 ,...,tn+1 ) = R t1 ,...,tn for each tn+1 ∈ T and Pt1 ,...,tn+1 (A1 × · · · × An × Xn+1 ) = n t ,...,t Pt1 ,...,tn (A1 × · · · × An ) for each A1 × · · · × An ∈ R t1 ,...,tn , where πt11 ,...,tn+1 : Xt1 × · · · × Xtn+1 → n Xt1 × · · · × Xtn is the natural projection, Al ⊂ Xtl for each l = 1, . . . , n. Suppose that this cylindrical distribution is bounded, that is, supt1 ,...,tn ∈T,n∈N kPt1 ,...,tn k < ∞ and there exists limt1 ,...,tn ∈T0 ;n∈N Pt1 ,...,tn (Xt1 × · · · × Xtn ) ∈ K, where T0 := {t ∈ T : Pt (Xt ) 6= 0}, T \ T0 is finite. 2.41. Theorem (the non-Archimedean analog of the Kolmogorov theorem). A cylindrical distribution Pt1 ,...,tn from § 2.40 has an extension to a bounded measure P on a comS pletion R P of R := t1 ,...,tn ∈T,n∈N G t1 ,...,tn relative to P, where G t1 ,...,tn := (πt1 ,...,tn )−1 (R t1 ,...,tn ) and πt1 ,...,tn : X → Xt1 × · · · × Xtn is the natural projection.

Non-Archimedean Valued Measures

2.3.

109

Quasi-invariant Ks -Valued Measures

In this section after few preliminary statements there are given the definition of a quasiinvariant measure and the theorems about quasi-invariance of measures relative to transformations of a Banach space X over K. S 3.1. Let X be a Banach space over K, (Ln : n) be a sequence of subspaces, cl( n Ln ) = j X, Ln ⊂ Ln+1 for each n, µ j be probability measures, µ2 ≪ µ1 , (µLn ) be sequences of weak distributions, also let there exist derivatives ρn (x) = µ2Ln (dx)/µ1Ln (dx) and the following limit ρ(x) := limn→∞ ρn (x) exists. Theorem. If µ j are Ks -valued and [ρn (PLn x) : n] converges uniformly on A f (X, µ1 )compact subsets in X, supn kρn kNµ1 < ∞, then this is equivalent to the following: ρ(x) = µ2 (dx)/µ1 (dx) ∈ L(µ1 ) and limn→∞ kρ(x) − ρn (PLn x)kNµ1 = 0. Proof. For each A ∈ Bco(L) the equality is accomplished: µ2L (A) =

Z

A

ρL (x)µ1L (dx) =

Z

PL−1 (A)

ρL (PL x)µ1 (dx).

Then for each ψ ∈ L(µ1 |PL−1 [Bco(L)]) we have Z Z

X

ψ(x)µ (dx) = 2

X

Z

X

ρL (PL x)ψ(x)µ1 (dx), consequently,

ρn+1 (x)ψ(x)µ1 (dx) =

Z

X

ψ(x)µ2 (dx) =

Z

X

ρn (x)ψ(x)µ1 (dx),

where ρLn = ρn , ψ ∈ L(µ1 |PL−1 [Bco(Ln+1 )]). From Theorem 2.33 and Definition 2.31 the n+1 statement follows. 3.2. Theorem. (A). Measures µ j : E → Ks , j = 1, 2, for a Banach space X over K are orthogonal µ1 ⊥ µ2 if and only if Nµ1 (x)Nµ2 (x) = 0 for each x ∈ X. (B). If for measures µ j : E → Ks on a Banach space X over K is satisfied ρ(x) = 0 for each x with Nµ1 (x) > 0, then µ1 ⊥ µ2 ; the same is true for a completely regular space X with ind(X) = 0 and ρ(x) = µ2 (dx)/µ1 (dx) = 0 for each x with Nµ1 (x) > 0. Proof. (A). From Definition 2.31 it follows that there exists F ∈ E with kX \ Fkµ1 = 0 and kFkµ2 = 0. Remind Theorem 7.20 about tight measures from [Roo78]. For a function f : X → K the conditions (α) and (β) are equivalent. (α). For every µ ∈ M(X) the function f is µ-integrable. (β). The function f is bounded and for each compact subset C in X the restriction of f to C is continuous. In view of Theorems 7.6 [Roo78] recalled above and the Theorem just cited above the characteristic function chF of the set F belongs to L(µ1 ) ∩ L(µ2 ) such that Nµ j (x) are semicontinuous from above, kchF kNµ2 = 0, kchX\F kNµ1 = 0, consequently, Nµ1 (x)Nµ2 (x) = 0 for each x ∈ X. Recall Lemma 7.2 [Roo78]. Let µ be a measure on R . There exists a unique function Nµ : X → [0, ∞) for which (Ri) kξU kNµ = kUkµ for each U ∈ R , where ξU denotes the characteristic function of U;

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(Rii) if φ : X → [0, ∞) and kξU kφ ≤ kUkµ for all U ∈ R , then φ ≤ Nµ , moreover, Nµ is given by the formula Nµ (x) = infU∈R ,x∈U kUkµ . On the other hand, if Nµ1 (x)Nµ2 (x) = 0 for each x, then for F := [x ∈ X : Nµ2 (x) = 0] due to the lemma recalled just above kFkµ2 = kchF kNµ2 = 0. Moreover, in view of Theorem T 7.6[Roo78] reminded above F = ∞ n=1 Us−n , where Uc := [x ∈ X : Nµ2 (x) < c] are open in X, hence chF ∈ L(µ1 ) ∩ L(µ2 ) and Nµ1 |(X\F) = 0, consequently, kX \ Fkµ1 = 0. (B). In view of Theorems 2.19 and 2.37 for each A ∈ PL−1 [E(Ln )] and m > n: n R 1 2 A ρm (x)µ (dx) = µ (A), then from limn→∞ kρ(x) − ρn (PLn x)kNµ1 = 0 and Conditions 2.1.(iiii) on µ2 Statement (B) follows. 3.3. Note. The Radon-Nikodym theorem is not valid for µ j with values in Ks , so not all theorems for real-valued measures may be transferred onto this case. Therefore, the definition of absolute continuity of measures was changed (see § 2.31 above and [Sch71]). 3.4. Theorem. Let measures µ j and ν j be with values in Ks on Bco(X j ) for a Banach space X j over K and µ = µ1 ⊗ µ2 , ν = ν1 ⊗ ν2 on X = X1 ⊗ X2 , therefore, the statement ν ≪ µ is equivalent to ν1 ≪ µ1 and ν2 ≪ µ2 , moreover, ν(dx)/µ(dx) = (ν1 (P1 dx)/µ1 (P1 dx))(ν2 (P2 dx)/µ2 (P2 dx)), where Pj : X → X j are projectors. Proof. Recall the non-Archimedean analog of the Fubini theorem (see also Theorem 7.15 [Roo78]). Let µ and ν be K-valued measures on separating covering rings R and S on X and Y correspondingly. The finite unions of sets A × B for A ∈ R and B ∈ S form a covering ring R ⊗ S of X ×Y . Moreover, (Ri) there exists a unique measure µ × ν on R ⊗ S so that (µ × ν)(A × B) = µ(A)ν(B), Nµ×ν (x, y) = Nµ (x)Nν (y); R R R (Rii) if f ∈ L(µ), g ∈ L(ν), then f ⊗Rg ∈ L(µ × ν) and f ⊗ gd(µ × ν) = f dµ gdν; (Riii) if f ∈ L(µ ×Rν), then y 7→R R f (x, y)dµ(x) is ν-almost everywhere defined νintegrable function and f d(µ × ν) = f (x, y)dµ(x)dν(y). This theorem follows from the formulated just above non-Archimedean analog of the Fubini theorem and making the modification of the proof of Theorem 3.2.3 of Chapter I above. 3.5. Theorem. The non-Archimedean analog of the Kakutani theorem. Let X = ∏∞j=1 X j be a product of completely regular spaces X j with ind(X j ) = 0 and probability N N measures µ j , ν j : E(X j ) → Ks , also let µ j ≪ ν j for each j, ν = ∞j=1 ν j , µ = ∞j=1 µ j are measures on E(X), ρ j (x) = µ j (dx)/ν j (dx) are continuous by x ∈ X j , ∏nj=1 ρ j (x j ) =: tn (x) converges uniformly on A f (X, µ)-compact subsets in X, β j := kρ j (x)kφ j , φ j (x) := Nν j (x) on X j . If ∏∞j=1 β j converges in (0, ∞) (or diverges to 0), then µ ≪ ν and qn (x) = ∏nj=1 ρ j (x j ) converges in L(X, ν, Ks ) to q(x) = ∏∞j=1 ρ j (x j ) = µ(dx)/ν(dx) (or µ ⊥ ν respectively), where x j ∈ X j , x ∈ X. Proof. The countable additivity of ν and µ follows from Theorem 2.19. Then β j = kρ j kφ j ≤ kρ j kNν j = kXkµ j = 1, since Nν j ≤ 1 for each x ∈ X j , hence ∏∞j=1 β j can not be divergent to ∞. If this product diverges to 0 then there exists a sequence εb := m(b) ∏ j=n(b) β j for which the series converges ∑∞ b=1 εb < ∞, where n(b) ≤ m(b). For Ab := [x : m(b)

m(b)

(∏ j=n(b) ρ j (x j )) ≥ 1] there are estimates kAb kν ≤ supx∈Ab [∏ j=n(b) |ρ j (x j )|φ j (x j )] ≤ εb , consequently, kAkν = 0 for A = lim sup(Ab : b → ∞), since 0 < ∑∞ b=1 εb < ∞.

Non-Archimedean Valued Measures For Bb := X \ Ab we have:    m(b) kBb kµ ≤ sup ∏ |1/ρ j (x j )|ψ(x j ) = x∈Bb j=n(b)

m(b)

∏

j=n(b)

111

 kρ j (x j )kφ j = εb

, where ψ j (x) = Nµ j (x), since µ j (dx j ) = ρ j (x j )ν j (dx j ) and Nµ j (x) = |ρ j (x j )|Nν j (x) due to continuity of ρ j (x j ) (for ρ j (x j ) = 0 we set |1/ρ j (x j )|ψ j (x j ) = 0, because ψ j (x j ) = 0 for such x j ), consequently, k lim sup(Bb : b → ∞)kµ = 0 and kAkµ ≥ k lim inf(Ab : b → ∞)kµ = 1. This means that µ ⊥ ν. Suppose that ∏∞j=1 β j converges to 0 < β < ∞, then β ≤ 1 (see above). Therefore from the non-Archimedean analog of the Lebesgue reminded above in this Chapter it follows that tn (x) converges in L(X, µ, Ks ), since |tn (x)| ≤ 1 for each x and n, at the same time each tn (x) converges uniformly on compact subsets in the topology generated by A f (X, µ). Then for each bounded continuous cylindrical function f : X → Ks we have Z

f (x)µ(dx) = X

= lim

Z

n→∞ X

Z

X

f (x1 , . . . , xn )tn (x) ⊗nj=1 ν j (dx j )

f (x)tn (x)ν(dx) =

Z

X

ρ(x)ν(dx).

Approximating arbitrary h ∈ L(X, µ, Ks ) by such f we get the equality Z

h(x)µ(dx) = X

Z

h(x)ρ(x)ν(dx), X

consequently, ρ(x) = µ(dx)/ν(dx). 3.6. Theorem. Let ν, µ, ν j , µ j be probability measures with values in Ks , X and X j be the same as in § 3.5 and µ ≪ ν, then µ j ≪ ν j for each j and ∏∞j=1 β j converges to β, ∞ > β > 0, where β j = kρ j kφ j , φ j (x) = Nν j (x). Proof. For Ks -valued measures from P−1 j (Bco(X j )) ⊂ Bco(X) it follows that µ j ≪ ν j for each j, since ∏∞ ρ (x ) = ρ(x) ∈ L(X, ν) and ρ j (x j ) ∈ L(X j , ν j ), where x j = 1 j j Pj x, Pj : X → X j are projectors. Then ρ(x) = limn→∞ ∏n1 ρ j (Pj x) and kρ(x)kNν = limn→∞ kρ j kNν j . Since Nν j ≤ 1, then φ j (x) ≤ Nν j (x) and for φ = Nν , consequently, kρ(x)kφ = limn→∞ ∏nj=1 kρ j kφ j ≤ kρkNν = 1 (due to the definition of the Tihonov topology in X [see also § 2.3[Eng86]] and definition of k ∗ kφ ). If kρkφ = 0, then kρkNν = 0 and by Theorem 3.2(B) this would mean that ν ⊥ µ or µ = 0, but µ 6= 0, hence β > 0. 3.7. Definition. Let X be a Banach space over K, Y be a completely regular space with ind(X) = 0, ν : Bco(Y ) → Ks , µy : Bco(X) → Ks for each y ∈ Y , suppose µy (A) ∈ L(Y, ν) for each A ∈ Bco(X), kY kν < ∞, supy∈Y kXkµy < ∞, a family (µy (An ) : n) is converging uniformly by y ∈ C on each A f (Y, ν)-compact subset C in Y for each given shrinking family of subsets (An : n) ⊂ X. Then we define: (i) µ(A) =

Z

µy (A)ν(dy).

Y

A measure µ is called mixed. Evidently, Condition 2.1(i) is fulfilled; (ii): kAkµ ≤ (supy∈Y kAkµy )kAkν < ∞; (iii) is carried out due to the non-Archimedean analog of the Lebesgue theorem, since lim µ(An ) =

n→∞

Z

Y

(lim µy (An ))ν(dy) = 0. n

112

Sergey V. Ludkovsky

We define measures π j by the formula: (ii) π (A ×C) = j

Z

µ j,y (A)ν j (dy),

C

where j = 1, 2 and µy, j together with ν j are defined as above µy and ν. 3.8. Theorem. Let µ j be Ks -valued measures and π j , X and Y be the same as in § 3.7. (A). If π2 ≪ π1 , then ν2 ≪ ν1 and µ2,y ≪ µ1,y (mod ν2 ). (B). If ν2 ≪ ν1 and µ2,y ≪ µ1,y (mod ν2 ) and a Bco(X × Y, π1 )-measurable ˜ x) = µ2,y (dx)/µ1,y (dx) ∈ L(X × Y, π1 ) exists, then π2 ≪ π1 and function ρ(y, 2 ˜ x). π (d(x, y))/π1 (d(x, y)) = (ν2 (dy)/ν1 (dy))ρ(y, Proof. (A). From the conditions imposed on µ j,y and ν j it follows that for each φ ∈ L(X ×Y, π j ) due to the non-Archimedean analog of the Fubini theorem recalled above the following equality is accomplished  Z Z Z j,y j φ(x, y)µ (dx) ν j (dy), φ(x, y)π (d(x, y)) = X×Y

Y

X

also ρ(y, x) = π2 (d(x, y))/π1 (d(x, y)) ∈ L(X ×Y, π1 ), hence  Z ρ(y, x)µ1,y (dx) ∈ L(Y, ν1 ). ν2 (dy)/ν1 (dy) = X

˜ x) may be defined for Further we modify the proof of Theorem I.3.1 above. Then ρ(y, R ˜ x) = ρ(y, x)/ X ρ(y, x)µ1,y (dx) ∈ L(X, µ1,y ). ν2 -almost all y by ρ(y, (B). Let A ∈ Bco(X) × Bco(Y ), Ay := [y : (x, y) ∈ A], then π j (A) = R j,y j If kAkπ1 = 0, then kAy kµ1,y Nν1 (y) = 0 for each y ∈ Y , conY µ (Ay )ν (dy). sequently, kAkπ2 = 0, since ν2 (dy)/ν1 (dy) ∈ L(ν1 ), µ2,y (dx)/µ1,y (dx) ∈ L(µ1,y ), ρ˜ ∈ L(X × Y, π1 ) and Conditions (i, ii) in § 3.7 are satisfied. From this it follows that π2 (d(x, y))/π1 (d(x, y)) ∈ L(X × Y, π1 ), since ν2 (dy)/ν1 (dy) ∈ L(X × Y, π1 ) with supy kXkµ j,y < ∞. 3.9. Definition. For a Banach space X over K an element a ∈ X is called an admissible shift of a measure µ with values in Ks , if µa ≪ µ, where µa (A) = µ(S−a A) for each A in E ⊃ Bco(X), Sa A := a + A, ρ(a, x) := ρµ (a, x) := µa (dx)/µ(dx), Mµ := [a ∈ X : µa ≪ µ] (see § § 2.1 and 2.31). 3.10. Properties of Mµ and ρ from § 3.9. I. The set Mµ is a semigroup by addition, ρ(a + b, x) = ρ(a, x)ρ(b, x − a) for each a, b ∈ Mµ . R R Proof. RFor each continuous bounded f : X → Ks : X f (x)µa+b (dx) = X f (x + a + R b)µ(dx) = X f (x + a)ρ(b, x)µ(dx) = X f (x)ρ(b, x − a)ρ(a, x)µ(dx), since kXkµ < ∞ and f (x)ρ(b, x − a) ∈ L(µ), consequently, ρ(b, x − a)ρ(a, x) = ρ(a + b, x) ∈ L(µ) as a function of x and µa+b ≪ µ. II. If a ∈ Mµ , ρ(a, x) 6= 0 (mod µ), then µa ∼ µ, −a ∈ Mµ and ρ(−a, x) = 1/ρ(a,Rx − a). R Proof. For each continuous bounded f : X → Ks : X f(x)µ(dx) = X f(x) R µ(dx) = X [µa (dx)/µ(dx)]−1 µa (dx), since kXkµ < ∞, hence µa ∼ µ. III. If ν ≪ µ and ν(dx)/µ(dx) = g(x), then Mµ ∩ Mν = Mµ ∩ [a : µ([x : g(x) = 0, g(x − a)ρµ (a, x) 6= 0]) = 0] and ρν (a, x) = [g(x − a)/g(x)]ρµ (a, x) (mod ν) for a ∈ Mµ ∩ Mν .

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Proof. For each continuous bounded function f : X → Ks : a ∈ Mµ and X f (x + a)ν(dx) f (x)g(x − a)ρµ R(a, x)µ(dx) such that µ([x : g(x) = 0, g(x − a)ρµ (a, x) 6= 0]) = 0 we have = R X X f (x + a)ν(dx) = X f (x)[g(x − a)ρµ (a, x)/g(x)]ν(dx), since kXkν + kXkµ < ∞, Nν (x) = infBco(X)⊃U∋x supy∈U [ |g(y)|Nµ (y)], consequently, a ∈ Mµ ∩ Mν . If a ∈ Mµ ∩ Mν , then R

Z

X

f (x)ρν (a, x)g(x)µ(dx) =

Z

X

f (x)g(x − a)ρµ (a, x)µ(dx),

consequently, ρν (a, x)g(x) = g(x − a)ρµ (a, x) (mod µ) and µ([x : g(x) = 0, g(x − a)ρµ (a, x) 6= 0]) = 0. IV. If ν ∼ µ, then Mν = Mµ . V. For µ with values in Ks and X = K m , m ∈ N a family Mµ with a distance function r(a, b) = kρ(a, x) − ρ(b, x)kNµ(x) is a complete pseudo-ultrametrizable space. Proof. Let (an ) ⊂ Mµ be a Cauchy sequence relative to r, then (an ) is bounded in X by k ∗ kX , since for lim j→∞ kan j k = ∞ and r(an j , an j+1 ) < p− j for f ∈ L(µ) with a compact support we have k f (x + an j ) − f (x + an1 )kNµ < 1/p. Then for f with k f (x + an1 )kNν > 1/2 and k f kNν = 1 we get a contradiction: lim j k f (x + an j )kNµ > 1/2 − 1/p ≥ 0. This is impossible because of compactness of supp( f ). Therefore, (an ) is bounded, consequently, there exists a subsequence (an j ) =: (b j ) weakly converging in X to b ∈ X. Therefore, R R θ j (z) = X χe (z(x + b j ))µ(dx) χe (z(b j ))θ(z) = X χe (z(x))ρ(b j , x)µ(dx), lim j z(b j ) = z(b) and lim j θ j (z) = χe (z(b))θ(z) for each z ∈ X ′ . From the basic theorem about the Fourier-Stieltjes transform reminded above in this chapter it follows that there is ρ ∈ L(µ) with lim j kρ(b j , x) − ρ(x)kNµ = 0, since L(µ) is the Banach space and µ j corresponding to θ j converges in the Banach space M(X). Therefore, Z

X

χe (z(x))µb (dx) =

Z

X

χe (z(x))ρ(x)µ(dx)

for each z ∈ X ′ = K m , consequently, ρ(x) = µb (dx)/µ(dx). 3.11. Definition. For a Banach space X over K and a measure µ : Bco(X) → Ks , a ∈ X, kak = 1, a vector a is called an admissible direction, if a ∈ MµK := [z : kzkX = 1, λz ∈ Mµ and ρ(λz, x) 6= 0 (mod µ) (relative to x) and for each λ ∈ K] ⊂ X. Let a ∈ MµK we denote by L1 := Ka, X1 = X ⊖ L1 , µ1 and µ˜ 1 are the projections of µ onto L1 and X1 respectively, µ˜ = µ1 ⊗ µ˜ 1 be a measure on Bco(X), given by the the following equation µ˜ (A ×C) = µ1 (A)˜µ1 (C) on Bco(L1 ) × Bco(X1 ) and extended on Bco(X), where A ∈ Bco(L1 ) and C ∈ Bco(X1 ). 3.12. Definition and notes. A measure µ : Bco(X) → Ks for a Banach space X over K is called a quasi-invariant measure if Mµ contains a K-linear manifold Jµ dense in X. From § 3.10 and Definition 3.11 it follows that Jµ ⊂ MµK . Let (e j : j ∈ N) be orthonormal basis in X, H = spanK (e j : j). We denote Ω(Y ) = [µ|µ is a measure with a finite total variation on Bco(X) and H ⊂ Jµ ], where Y = Ks . 3.13. Theorem. If µ : B f (Y ) → F is a σ-finite measure on Bco(Y ), Y is a complete separable ultra-metrizable K-linear subspace such that co(S) is nowhere dense in Y for each compact S ⊂ Y , where K and F are infinite non-discrete non-Archimedean fields with multiplicative ultra-norms | ∗ |K and | ∗ |F . Then from Jµ = Y it follows that µ = 0. S Proof. Since µ is σ-finite, then there are (Y j : j ∈ H) ⊂ Bco(Y ) such that Y = j∈H Y j and 0 < kµ|Bco(Y j )k ≤ 1 for each j, where H ⊂ N, Y j ∩Yl = 0/ for each j 6= l. If card(H) =

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ℵ0 , then we define a function f (x) = s j /kY j kµ for µ with values in F, where s is fixed R with 0 < |s|F < 1, s ∈ N). Then we define a measure ν(A) = A f (x)µ(dx), A ∈ Bco(Y ). Therefore, kY kν ≤ 1 and Jν = Y , since f ∈ L(Y, µ, F). Hence it is sufficient to consider µ with kµk ≤ 1 and µ(Y ) = 1. For each n ∈ N in view of the Radonian property of Y there exists a compact Xn ⊂ Y such that kY \ Xn kµ < s−n . In Y there is a countable everywhere dense S subset (x j : j ∈ N), hence Y = j∈N B(Y, x j , rl ) for each rl > 0, where B(Y, x, rl ) = [y ∈ Y : d(x, y) ≤ rl ], d is an ultra-metric in Y , i.e. d(x, z) ≤ max(d(x, y), d(y, z)), d(x, z) = d(z, x), d(x, x) = 0, d(x, y) > 0 for x 6= y for each x, y, z ∈ Y . Therefore, for each rl = 1/l, l ∈ N there exists k(l) ∈ N such that kY \ Xn,l kµ < s−n−l due to compactness of Yc = [y ∈ Y : Sk(l) Nµ (y) ≥ c] for each c > 0, where Xn,l := j=1 B(Y, x j , rl ), consequently, kY \ Xn kµ ≤ s−n for T∞ Xn := l=1 Xn,l . The subsets Xn are compact, since Xn are closed in Y and the metric d on Xn is completely bounded and Y is complete (see also Theorems 3.1.2 and 4.3.29 [Eng86]). S Then 0 < kXkµ ≤ 1 for kY \ Xkµ = 0 and for X := spanK ( ∞ Xn ). Sn=1 n ˜ The sets Yn := co(Yn ) are nowhere dense in Y for Yn = l=1 Xl , consequently, spanKYn S / is dense in Y due to the Baire category are nowhere dense in Y . Moreover, (Y \ ∞ n=1 Yn ) 6= 0 theorem (see also § § 3.9.3 and 4.3.26 [Eng86]). Therefore, y + X ⊂ Y \ X for y ∈ Y \ X and from Jµ = Y it follows that kXkµ = 0, since ky + Xkµ = 0 (see § § 2.32 and 3.12 above). Hence we get the contradiction, consequently, µ = 0. 3.14. Corollary. If Y is a Banach space or a complete countably-ultra-normable infinite-dimensional over K space, µ : Bco(Y ) → Ks , K and F are the same as in § 3.13 and Jµ = Y , then µ = 0. Proof. The space Y is evidently complete and ultra-metrizable, since its topology is given by a countable family of ultra-norms. Recall the theorem about hulls of compact sets (see also § (5.7.5) in [NB85]). (NBi). In any topological vector space X the balanced hull Hb of a bounded or compact set H is again totally bounded or compact respectively. (NBii). If H is a totally bounded subset of a locally convex space X, then so is its convex hull Hc and therefore its disked hull Hbc . (NBiii). If H is a compact subset of a locally convex space X, then its convex hull Hc and disked hull Hbc are compact if and only if Hc and Hbc are complete respectively. If X is complete, then clX Hc and clX Hbc are compact. Due to this theorem co(S) is nowhere dense in Y for each compact S in Y , since co(S) = cl(Sbc ) is compact in Y and does not contain in itself any open subset in Y , since Y is infinite dimensional over the field K. 3.15. Theorem. Let X be a separable Banach space over a locally compact infinite field K with a nontrivial normalization such that either K ⊃ Qp or char(K) = p > 0. Then there are probability measures µ on X with values in Ks (s 6= p) such that µ are quasiinvariant relative to a dense K-linear subspace Jµ . Proof. Let S( j, n) := p j B(K, 0, 1) \ p j+1 B(K, 0, 1) for j ∈ Z and j ≤ n, S(n, n) := pn B(K, 0, 1), w′ be the Haar measure on K considered as the additive group (see also [Bou63-69, HR79, Roo78]) with values in Ks for s 6= p. Then for each c > 0 and n ∈ N there are measures m on B f (K) such that m(dx) = f (x)v(dx), | f (x) |> 0 for each x ∈ K and | m(pn B(K, 0, 1)) − 1 |< c, m(K) = 1, | m | (E) ≤ 1 for each E ∈ Bco(K) , where v = w′ , v(B(K, 0, 1)) = 1. Moreover, we can choose f such that a density ma (dx)/m(dx) =: d(m; a, x) be continuous by (a, x) ∈ K2 and for each c′ > 0, x and

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115

| a |≤ p−n : | d(m; a, x) − 1 |< c′ . Let f |S( j,n) := a( j, n) be locally constant, for example, a( j, n) = (1 − s)(1 − 1/p)s2n−1− j p−n for j < n, a(n, n) = (1 − s−n )p−n . Then taking f + h and using h(x) with 0 < supx |h(x)/ f (x)| = c” ≤ 1/sn we get | ya (dx)/y(dx) |=| ma (dx)/m(dx) |, where y(dx) = ( f + h)(x)m(dx). More generally it is possible to take g ∈ L(K, Bco(K), w′ , Ks ) such that g(x) 6= 0 for w′ -almost every x ∈ K and kgk = 1 and ∏∞ n=1 βn > 0 converges for each y = {yn : yn ∈ K, n ∈ N} in a proper dense subspace J in X = c0 , where gn (x) := g(x/an ), limn→∞ an = 0, 0 6= an ∈ K for each n ∈ N, βn := kρn kφn , ρn (x) := µn (dx)/νn (dx), φn (x) := Nλn (x), λn (dx) := gn (x)w′ (dx/an ), then use Theorem 3.5 for the measure νn (dx) := gn (x)w′ (dx/an ) and its shifted measure µn (dx) := νn (−yn + dx). Let {m( j; dx)} be a family of measures on K with the corresponding sequence {k( j)} such that k( j) ≤ k( j + 1) for each j and limi→∞ k(i) = ∞, where m( j; dx) corresponds to the partition [S(i, k( j))]. The Banach space X is isomorphic with c0 (ω0 , K), since K is spherically complete. It has the orthonormal basis {e j : j = 1, 2, . . . } and the projectors Pj x = (x(1), . . . , x( j)) onto Kj , where x = x(1)e1 + x(2)e2 + · · · . Then there exists a cylindrical measure µ generated by a consistent family of measures y( j, B) = b( j, E) j for B = P−1 j E and E ∈ B f (K ) [Bou63-69, DF91] where b( j, dz) = ⊗[m( j; dz(i)) : i = 1, . . . , j], z = (z(1), . . . , z( j)). Let L := L(t,t(1), . . . ,t(l); l) := {x : x ∈ X and | x(i) |≤ pa , a = −t − t(i) for i = 1, . . . , l, and a = −k( j) for j > l}, then L is compact in X, since X is Lindel¨of and L is sequentially compact (see also [Eng86]). Therefore, for each c > 0 there exists L such that kX \ Lkµ < c due to the choice of a( j, n). In view of the non-Archimedean analog of the Prohorov theorem for measures with values in Ks reminded above (see also 7.6(ii)[Roo78]) and due to Lemma 2.3 µ has the countably-additive extension on B f (X), consequently, also on the complete σ-field A f (X, µ) and µ is the Radon measure. Let z′ ∈ spanK {e j : j = 1, 2, . . . } and z” = {z( j) : z( j) = 0 for j ≤ l and z( j) ∈ S(n, n), j = 1, 2, . . . , n = k( j)}, l ∈ N, z = z′ + z”. Now take the restriction of µ on Bco(X). In view of Theorems 2.19, 3.5 above and also the lemma about isometric mappings I.1.4 [Roo78] recalled above (see also § I.3.20 above) there are m( j; dz( j) such that ρµ (z, x) = ∏{d( j; z( j), x( j)) : j = 1, 2, . . . } = µz (dx)/µ(dx) ∈ L((X, µ, Bco(X)), Ks ) for each such z and x ∈ X, where d( j; ∗, ∗) = d(m( j; ∗), ∗, ∗) and µz (X) = µ(X) = 1. 3.15.1. Theorem. Let X be a Banach space of separable type over a locally compact non-Archimedean field K and J be a dense proper K-linear subspace in X such that the embedding operator T : J ֒→ X is compact and nondegenerate, ker(T ) = {0}. Then a set M (X, J) of probability Ks -valued measures µ on Bco(X) quasi-invariant relative to J is of cardinality card(Ks )c . If J ′ , J ′ ⊂ J, is also a dense K-linear subspace in X, then M (X, J ′ ) ⊃ M (X, J). Proof. As in the proof of Theorem 3.20.1 choose for a given compact operator T an orthonormal base in X in which T is diagional and X is isomorphic with c0 over K such that in its standard base {e j : j ∈ N} the operator T is such that Te j = a j e j , 0 6= a j ∈ K for each j ∈ N, lim j→∞ a j = 0 (see Appendix). As in Theorem 3.15 take gn ∈ L(K, Bco(K), w′ (dx/an ), Ks ), gn (x) 6= 0 for v-a.e. x ∈ K and kgn k = 1 for each n, m ′ for which converges ∏∞ n=1 βn > 0 for each y ∈ J and such that ∏n=1 gn (xn )w (dxn /an ) =: n νLn (dx ) satisfies conditions of Lemma 2.3, where βn := kρn kφn , 0 6= an ∈ K for each n ∈ N, ρn (x) := µn (dx)/νn (dx), φn (x) := Nλn (x), λn (dx) := gn (x)w′ (dx/an ), then use Theorem 3.5 for the measure νn (dx) := gn (x)w′ (dx/an ) and µn (dx) := νn (−yn + dx), xn := (x1 , . . . , xn ),

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x1 , . . . , xn ∈ K for each n ∈ N. The family of such sequences of functions {gn : n ∈ N} has the cardinality card(Ks )c , since in L(ν) the subspace of step functions is dense and card(Bco(X)) = c. The family of all {gn : n} satisfying conditions above for J also satisfies such conditions for J ′ . From which the latter statement of this theorem follows. 3.16. Note. For a given m = w′ (see above) new suitable measures may be constructed, if to use images of measures mg (E) = m(g−1 (E)) such that for a diffeomor−1 phism g ∈ Di f f 1 (K) (see § A.3) we have mg (dx)/m(dx) =| (g′ (g−1 (x)) |K , where | ∗ |K = modK (∗) is the modular function of the field K associated with the Haar measure on K, at the same time | ∗ |K is the multiplicative norm in K consistent with its uniformity [Wei73]. Indeed, for K and X = Kj with j ∈ N and the Haar measure v = w′ on X, vX := v with values in Ks for s 6= p and for a function f ∈ L(X, v, Ks ) we have: R R ′ f (x)v(dx) = A f (g(y))|g (y)|K v(dy), where modK (λ)v(dx) := v(λdx), λ ∈ K, since g(A) v(B(X, 0, pn )) ∈ Q, Nv (x) = 1 for each x ∈ X, consequently, from fk → f in L(g(A), v, Ks ) whilst k → ∞ it follows that fk (g(x)) → f (g(x)) in L(A, v, Ks ), where fk are locally constant, A is compact and open in X. Henceforward, quasi-invariant measure µ on Bco(c0 (ω0 , K)) constructed with the help of projective limits or sequences of weak distributions of probability measures (µH(n) : n) are considered, for example, as in Theorem 3.15 such that (i) µH(n) (dx) = fH(n) (x)vH(n) (dx), dimK H(n) = m(n) < ℵ0 for each n ∈ N, where S fH(n) ∈ L(H(n), vH(n) , Ks ), H(n) ⊂ H(n + 1) ⊂ · · · , cl( n H(n) = c0 (ω0 , K), if it is not specified in another manner. In accordance with the Schikhof’s Theorem 8.9 [Roo78]: if G is a zero-dimensional topological group with unit element e, ν is a tight measure, ν ∈ M(G), ν 6= 0, then the following conditions are equivalent: (α) the mapping s 7→ νs is continuous G → M(G); (β) this mapping s 7→ νs is continuous at e; (γ) G has a left Haar measure h and ν ∈ L(G), where L(G) denotes the set of tight absolutely continuous measures relative to the Haar measure h on G. Each ν ∈ L(G) can be written in the form ν = f h, where f ∈ C∞0 (G) is a continuous function so that for each b > 0 there exists a compact subset Eb in G with | f | < b outside a compact subset Eb . For probability quasi-invariant measure with values in Ks , if shifts x 7→ x+y by y ∈ H(n) are continuous from H(n) to M(H(n)) (see § 2.1), that is, y → µyH(n) , where µH(n) (y + A) =: µyH(n) (A) for A ∈ Bco(H(n)), then due to the Schikhof’s theorem about tight measures on zero-dimensional groups µH(n) satisfies (i). As will be seen below such measures µ are quasi-invariant relative to families of the cardinality c = card(R) of linear and non-linear transformations U : X → X. Moreover, for each V open in X we have kV kµ > 0, when fH(n) (x) 6= 0 for each n ∈ N and x ∈ H(n). Let µ be a probability quasi-invariant measure satisfying (i) and (e j : j) be orthonormal basis in Mµ , H(n) := spanK (e1 , . . . , en ), we denote by ˆ x) = limn→∞ ρn (Pn a, Pn x), ρˆ µ (a, x) = ρ(a, n ρ (Pn a, Pn x) := fH(n) (Pn (x − a))/ fH(n) (Pn x) for each a and x for which this limit exists ˆ x) = 0 in the contrary case, where Pn : X → H(n) are chosen consistent projecand ρ(a, ˆ x), if µa (X) = µ(X) and ρ(a, ˆ x) ∈ L(X, µ, Ks ) as a function by x tors. Let ρ(a, x) = ρ(a, ˆ x)µ(dx), ρ(a, x) is not defined when µa (X) = µ(X) or and kXkNν = 1, where ν(dx) := ρ(a,

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kXkNν 6= 1, this condition of the equality to 1 may be satisfied, for example, for continuˆ x) ∈ L(µ) by x for each given a, if limn ρn (a, x) converges ous fH(n) with continuous ρ(a, uniformly by x. If for some another basis (e˜ j : j) and ρ˜ is accomplished (ii) kX \ Skµ = 0, then ρ(a, x) is called regularly dependent from a basis, where S := T ˜ x)]). a∈Mµ [x : ρ(a, x) = ρ(a, 3.17. Lemma. Let µ be a probability measure, µ : Bco(X) → Ks , X be a Banach space over K, suppose that for each basis (e˜ j : j) in Mµ a quasi-invariance factor ρ˜ satisfies the following conditions: ˜ j , x), j = 1, . . . , N, are defined for a given x ∈ X and for each λ j ∈ K then a (1) if ρ(a ˜ ∑nj=1 λ j a j , x) is continuous by λ j , j = 1, . . . , N; function ρ( S (2) there exists an increasing sequence of subspaces H(n) ⊂ Mµ , cl( n H(n)) = ˜ n a, x) X, with projectors Pn : X → H(n), B ∈ B f (X), kBkµ = 0 such that limn→∞ ρ(P ˜ = ρ(a, x) for each a ∈ Mµ and x ∈ / B for which is defined ρ(a, x). Then ρ(a, x) depends regularly from the basis. S Proof. There exists a subset S dense in each H(n), hence kB′ kµ = 0 for B′ = a∈S [x : ˜ x)]. From (1) it follows that ρ(a, ˜ x) = ρ(a, x) on each H(n) for x ∈ ρ(a, x) 6= ρ(a, / B′ . From ˜ x) for each a ∈ Mµ and x ∈ X \ (B′ ∪ spanK S ⊃ H(n) and (2) it follows that ρ(a, x) = ρ(a, B), consequently, Condition 3.16(ii) is satisfied, since from ρ(a, x) ∈ L(µ) it follows that ˜ x) ∈ L(µ) as the function by x. ρ(a, 3.18. Lemma. If a probability quasi-invariant measure µ : Bco(X) → Ks satisfies Condition 3.16(i), then there exists a compact operator T : X → X such that Mµ ⊂ (T X)∼ , where X is the Banach space over K. Proof. Products of tight measures are tight measures (see also Theorem 7.28 [Roo78]), Nm(n) m(n) whence for µH(n) (dx) = j=1 µKe( j) (dx j ) is accomplished NµH(n) (x) = ∏ j=1 NµKe( j) (x j ), where x = (x1 , .., xm(n) ), x j ∈ K. From Theorem 7.6[Roo78] formulated above and Lemma I.2.5 it follows that for each 1 > c > 0 there are R j = R j (c) with [x j : NµKe( j) (x j ) ≥ c] ⊂ B(K, 0, R j ) and lim j→∞ R j = 0. Choosing c = c(n) = s−n , n ∈ N and using ∏∞j=1 = ε j = 0 whilst 0 < ε j < c < 1 for each j we get that there exists a sequence [r j : j] for which card[ j : |a j | > r j ] < ℵ0 for each a ∈ Mµ , since [x ∈ X : |x j | ≤ r j for all j] is a compact subgroup in X, where a = (a j : j), a j ∈ K, r j > 0, lim j r j = 0. Therefore, Mµ ⊂ (T X)∼ for T = diag(T j : j) and |T j | ≥ r j for j ∈ N. 3.19. Let X be a Banach space over K, | ∗ |K = modK (∗), U : X → X be an invertible linear operator, µ : Bco(X) → Ks be a probability quasi-invariant measure. The uniform convergence of a (transfinite) sequence of functions on A f (V, ν)-compact subsets of a topological space V is called the Egorov condition, where ν is a measure on V . ˜ x)), where dom( f ) Theorem. Let pairs (x − Ux, x) and (x − U −1 , x) be in dom(ρ(a, −1 ˜ ˜ denotes a domain of a function f , ρ(x − Ux, x) 6= 0, ρ(x − U x, x) 6= 0 (mod µ) and µ ˜ P˜n (x −Ux), x) =: ρ˜ n (x) ∈ L(µ) and ρ( ˜ P˜n (x −U −1 x, x) =: satisfies Condition 3.16(i), also ρ( ρ¯ n (x) ∈ L(µ) converge uniformly on A f (X, µ)-compact subsets in X such that there exists g ∈ L(µ) with |ρ˜ n (x)| ≤ |g(x)| and |ρ¯ n (x)| ≤ |g(x)| for each x ∈ X and each projectors S ˜ ˜ ˜ + 1) ⊂ . . . , that is, Egorov conditions ˜ = X, H(n) ⊂ H(n P˜n : X → H(n) with cl( n H(n)) ˜ ¯ are satisfied for ρn and ρn . Then ν ∼ µ and ˜ −U −1 x, x), (i) ν(dx)/µ(dx) = |det(U)|K ρ(x

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if ρ depends regularly from the base, then ρ˜ may be substituted by ρ in formula (i), where ν(A) := µ(U −1 A) for each A ∈ Bco(X). Proof. In view of Lemma 3.18 there exists a compact operator T : X → X such that Mµ ⊂ (T X)∼ , consequently, (U − I) is a compact operator, where I is the identity operator. From the invertibility of U it follows that (U −1 − I) is also compact, moreover, there exists ˜ det(U) ∈ K. Let g be a continuous bounded function, g : H(n) → Ks , whence Z

X

φ(x)ν(dx) =

Z

˜ H(n)

g(x)[ fH(n) (U −1 x)/ fH(n) (x)]|det(Un )|K µH(n) (dx), ˜ ˜ ˜

˜ ˜ ˜ for φ(x) = g(P˜n x), where subspaces exist such that H(n) ⊂ X, (U −1 − I)H(n) ⊂ H(n), S ˜ ˜ ˜ = X, Un := rˆn (U), rn = P˜n : X → H(n) (see § § I.3.8 and II.3.16), H(n) ⊂ cl( n H(n)) ˜ H(n + 1) ⊂ · · · due to compactness of (U − I). In view of the non-Archimedean analog of the Lebesgue convergence theorem due to fulfillment of the Egorov conditions for ρ˜ n and ρ¯ n (see also §7.6 [MS63] or § 7.F [Roo78]) ˜ −U −1 x, x) ∈ L(µ), where Jm = Jm,ρ , since ρ(x Jm := Jm,ρ :=

Z

X

Z

X

g(P˜m x)ν(dx) and

˜ −U −1 x, x)|det(U)|K µ(dx). g(P˜m x)ρ(x

Indeed, there exists n0 such that |u(i, j) − δi, j | ≤ 1/p for each i and j > n0 , consequently, |det(Un )|K = |det(U)|K for each n > n0 . Then due to Condition 3.16.(i) and the Egorov conditions (see also § 3.3) there exists lim [µH(n) (d P˜n x)/νH(n) (d P˜n x)] = µ(dx)/ν(dx) (mod ν). ˜ ˜

n→∞

Further analogously to the proof of Theorem I.3.24 above. 3.20. Let X be a Banach space over K, | ∗ |K = modK (∗) with a probability quasiinvariant measure µ : Bco(X) → Ks and Condition 3.16(i) be satisfied, also let U fulfils the following conditions: (i) U(x) and U −1 (x) ∈ C1 (X, X) (see also § A.3); (ii) (U ′ (x) − I) is compact for each x ∈ X; (iii) (x −U −1 (x)) and (x −U(x)) ∈ Jµ for µ − a.e. x ∈ X; (iv) for µ-a.e. x pairs (x −U(x); x) and (x −U −1 (x); x) are contained in a domain of ρ(z, x) such that ρ(x −U −1 (x), x) 6= 0, ρ(x −U(x), x) 6= 0 (mod µ); (v) kX \ S′ kµ = 0, where S′ := ([x : ρ(z, x) is defined and continuous by z ∈ L]) for each finite-dimensional L ⊂ Jµ ; (vi) there exists S with kSkµ = 0 and for each

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x ∈ X \ S and for each z for which there exists ρ(z, x) satisfying the following condition: limn→∞ ρ(Pn z, x) = ρ(z, x) and the convergence is uniform for each finite-dimensional L ⊂ Jµ by z in L ∩ [x ∈ Jµ : | x |≤ c], where c > 0, Pn : X → H(n) are projectors onto finitedimensional subspaces H(n) over K such that H(n) ⊂ H(n + 1) for each n ∈ N and cl ∪ {H(n) : n} = X; (vii) there exists n for which for all j > n and x ∈ X mappings V ( j, x) := x + Pj (U −1 (x) − x) and U( j, x) := x + Pj (U(x) − x) are invertible and lim j | detU ′ ( j, x) |=| detU ′ (x) |, lim j | detV ′ ( j, x) |= 1/ | detU ′ (x) | with the Egorov convergence in (vi) by z for ρ(Pn z, x) and in (vii) by x for |det(U ′ ( j, x))| and |det(V ′ ( j, x))| for µ with values in Ks . Theorem. The measure ν(A) := µ(U −1 (A)) is equivalent to µ and (i) ν(dx)/µ(dx) =| detU ′ (U −1 (x)) |K ρ(x −U −1 (x), x). Proof. The beginning of the proof is analogous to that of § I.3.25. Due to Conditions (vi, vii) we get limn ρ(x − Vn−1 x, x) = ρ(x − U1−1 x, x) in L(µ) due to the Egorov conditions. Then J1 = J1,ρ due to the Lebesgue convergence theorem, where J1 = J1,ρ :=

Z

X

Z

X

f (U1 x)µ(dx),

f (x)ρ(x −U1−1 x, x)|detU1 |K µ(dx)

for continuous bounded function f : X → Ks . Analogously for U1−1 instead of U1 . Using ¯ −1 x) := f (x)ρµ (x −U −1 x, x) and Properties 3.10 we get that instead of f the function Φ(U 1 1 ρµ (U1 x − x,U1 x)ρµ (x − U1 x, x) = 1 (mod µ). Therefore, for U = U1U2 with diagonal U1 and upper triangular U2 and lower triangular U3 operators with finite-dimensional over K R f (Ux)µ(dx) subspaces (U − I)X, j = 1, 2, 3, the following equation is accomplished j X R = X f (x)ρµ (x − U −1 x, x)|detU|K µ(dx). If either (S−1U − I)X = L or (U −1 S − I)X = L, then from the decomposition given in (I) U = SU2U1U3 , we have either (U j − I)X = L or (U j−1 − I)X = L respectively, j = 1, 2, 3 due to formulas from § A.1, since corresponding non-major minors are equal to zero. If U is an arbitrary linear operator satisfying the conditions of this theorem, then from (iv − vi) and (I, II) for each continuous bounded function f : X → Ks we have J = Jρ , where J := Jρ :=

Z

X

Z

f (U(x))µ(dx) and X

f (x)ρµ (x −U −1 (x), x)|detU|K µ(dx).

Analogously for U −1 , moreover, ρ(x − U −1 (x), x)|detU|K =: h(x) ∈ L(µ), h(x) 6= 0 (mod µ), since there exists detU. Suppose U is polygonal (see § I.3.25). Then U −1 is also polygonal, U ′ (x) = V ( j) for R R x ∈ Y ( j) and X f (a(i) +V (i)x)µ(dx) = f (a(i) + x)ρµ (x −V −1 (i)x, x) ×|det(V (i))|K µ(dx) for each continuous bounded f : X → Ks and each i. From a( j) ∈ Mµ and § 3.10 we get

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f (a( j) +V ( j)x)µ(dx) = X f (x)ρ(x −V ( j)−1 (x − a( j)), x)|detV ( j)|K µ(dx). Let Hk, j := [x ∈ X : V (k)−1 x = V ( j)−1 x], assume without loss of generality that V (k) 6= V ( j) or a(k) 6= a( j) for each k 6= j, since Y (k) 6= Y ( j) (otherwise they may be united). Therefore, Hk, j 6= X. If kHk, j kµ > 0, then from X ⊖ Hk, j ⊃ K it follows that Mµ ⊂ Hk, j , but cl(Hk, j ) = Hk, j and −1 cl(Mµ ) = X. This contradiction means that kAk µ = 0, where A = [x : V (k) (x − a(k)) = R R V ( j)−1 (x − a( j))]. Then X f (U(x))µ(dx) = X f (x)ρ(x −U −1 (x), x) |detU ′ (x)|−1 K µ(dx). Then as in § I.3.25(V ) for the construction of the sequence {U( j, ∗) : j} it is sufficient to construct a sequence of polygonal functions {a(i, j; x)}, that is a(i, j; x) = lk (i, j)(x) + ak for x ∈ Y (k), where lk (i, j) are linear functionals, ak ∈ K, Y (k) are closed in X, Int(Y ( j)) ∩ S Int(Y (k)) = 0/ for each k 6= j, m k=1 Y (k) = X, m < ℵ0 . For each c > 0 there exists Vc ⊂ ¯ 1 s(i, j; ∗))(x, e(k),t) are equiuniformly X with kX \ Vc k < c, the functions s(i, j; x) and (Φ continuous (by x ∈ Vc and by i, j, k ∈ N) on Vc . Choosing c = c(n) = s−n and using δ-nets in Vc we get a sequence of polygonal mappings (Wn : n) converging by its matrix elements by Egorov in the Banach space L(X, µ, Ks ), from Condition (i) it follows that it may be chosen equicontinuous for matrix elements s(i, j; x), ds(i, j; x) and s(i, Pj x) by i, j (the same is true for U −1 ). Then calculating integrals as above for Wn with functions f , using the Lebesgue convergence theorem we get the equalities analogous to written in § I.3.25(III) for J and Jρ of the general form. From ν(dx)/µ(dx) 6= 0 (mod µ) and § 2.19 we get the statement of the theorem. 3.21. Examples. Let X be a Banach space over the field K with the normalization group ΓK = ΓQp . We consider a diagonal compact operator T = diag(t j : j ∈ N) in a fixed orthonormal basis (e j : j) in X such that kerT := T −1 0 = {0}. Let 0 ν′j (dx j ) = C′ (ξ j )s−q min(0,ord p ((x j −x j )/ξ j )) v(dx j ) for the Haar measure v : Bco(K) → Qs , then ν′j (B f (K)) ⊂ Cs . We choose constant functions C′ (ξ j ) such that ν′j be a probability measure, where x0 = (x0j : j) ∈ X, x = (x j : j) ∈ X, x j ∈ K. N With the help of products j ν′j (dx j ) as in § 3.15 we can construct a probability quasiinvariant measure µT on X with values in Cs , since cl(T X) is compact in X and spanK (e j : T j) =: H ⊂ Jµ . From λ∈B(K,0,1)\0 cl(λT X) = {0} we may infer that for each c > 0 there T exists a compact Vc (λ) ⊂ X such that kX \Vc (λ)kµ < c and λ6=0 Vc (λ) = {0}, consequently, R

X

R

lim

Z

|λ|→0 X

f (x)µλT (dx) = f (0) = δ0 ( f ),

hence µλT is weakly converging to δ0 whilst |λ| → 0 for the space of bounded continuous functions f : X → Cs . q From Theorem 3.6 we conclude that from ∑∞j=1 |y j /ξ j | p < ∞ it follows y ∈ JµT . Then q for a linear transformation U : X → X from ∑ j |e˜ j (x − U(x))/ξ j | p < ∞ it follows that x − U(x) ∈ Jµ and a pair (x − U(x), x) ∈ dom(ρ(a, z)). Moreover, for ρ corresponding to µT conditions (v) and (vi) in § 3.20 are satisfied. Therefore, for such y and S ∈ A f (X, µ) a quantity |µ(ty + S) − µ(S)| is of order of smallness |t|q whilst t → 0, hence they are pseudodifferentiable of order b for 0 < Re(b) < q (see also § 4 below). It is interesting also to discuss a way of solution of one problem formulated in [KE92] that there does not exist a σ-additive Qp -valued measure with values in X over Qp such that it would be an analog of the classical Gaussian measure. In the classical case this

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121

means in particular a quasi-invariance of a measure relative to shifts on vectors from a dense subspace. We will show, that on a Banach space X over K ⊃ Qp for each prime number p there is not a σ-additive µ 6= 0 with values in Kp such that it is quasi-invariant relative to shifts from a dense subspace. Details can be lightly extracted from the results given above. Let on (X, Bco(X)) there exists such µ. With the help of suitable compact operators a cylindrical measure on an algebra of cylindrical subsets of X generates quasiinvariant measures, so we can suppose that µ is quasi-invariant. Then it produces a sequence of a finite-dimensional distribution {µLn : n ∈ N} analogously to § 2 and § 3, where Ln are subspaces of X with dimensions over K equal to n. Each measure µLn is σ-additive. From the quasi-invariance of µ it follows, that Ln can be chosen such that µLn are quasiinvariant relative to the entire Ln . But in view of Chapters 7-9 [Roo78] and [Sch84] for measures with values in Kp (see also Proposition 11 from § VII.1.9 [Bou63-69]) this means that µLn is equivalent to the Haar measure on Ln with values in Kp . The space Ln as the additive group can be considered over Qp , moreover, for each continuous linear functional φ : Kp → Qp considered as the finite-dimensional Banach space over Qp the measure φ ◦ µLn (∗) is non-trivial for some φ. Consequently, on Ln there would be the Haar measure with values in Qp , but this is impossible due to Chapter 9 in [Roo78], since Ln is not the p-free group. We get the contradiction, that is, such µ does not exist. 3.22. Theorem. Let A be a complete normed algebra over the local field K. If a nontrivial Ks -valued measure µ on Bco(A) is quasi-invariant relative to dense subalgebra A′ (relative to linear shifts and left (or right) multiplication), then A is finite dimensional over K. 3.23. Theorem. Let A be a Banach space over the local field K. If µ is a non-trivial Ks valued measure on Bco(A) quasi-invariant relative to shifts from a dense K-linear subspace L′ , then there exists a nontrivial topological group G of K-linear automorphisms of A such that µ is also quasi-invariant relative to G. Proof. These statements of 3.22, 3.23 follow from Theorem 3.19 and Lemma 3.18 analogously to § § I.3.28, I.3.29 above.

2.4.

Pseudo-differentiable Ks -Valued Measures

4.1. Note and Definition. A function f : K → Us is called pseudo-differentiable of order b, if there exists the following integral: PD(b, f (x)) :=

Z

[( f (x) − f (y)) × g(x, y, b)]dv(y). K

We introduce the following notation PDc (b, f (x)) for such integral by B(K, 0, 1) instead of the entire K. Where g(x, y, b) := s(−1−b)×ord p (x−y) with the corresponding Haar measure v with values in Ks , where b ∈ Cs and |x|K = p−ord p (x) , Cs denotes the field of complex numbers with the non-Archimedean normalization extending that of Qs , Us is a spherically complete field with a normalization group ΓUs := {|x| : 0 6= x ∈ Us } = (0, ∞) ⊂ R such that Cs ⊂ Us , 0 < s is a prime number (see also [Dia84, Roo78, Sch84, Wei73]). For each γ ∈ (0, ∞) there exists α = logs (γ) ∈ R, ΓUs = (0, ∞), hence sα ∈ Us is defined for each α ∈ R, where logs (γ) = ln(γ)/ln(s), ln : (0, ∞) → R is the natural logarithmic function such that ln(e) = 1. The function sα+iβ =: ξ(α, β) with α and β ∈ R is defined due to

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Sergey V. Ludkovsky

the algebraic isomorphism of Cs with C (see also [Kob77]) in the following manner. Put sα+iβ := sα (si )β and choose as si a marked number in Us such that si := (EXPs (i))ln s , where + EXPs : Cs → C+ s is the exponential function, Cs := {x ∈ Cs : |x − 1|s < 1} (see Proposition 45.6 [Sch84]). Therefore, |EXPs (i) − 1|s < 1, hence |EXPs (i)|s = 1 and inevitably |si |s = 1. Therefore, |sα+iβ |s = s−α for each α and β ∈ R, where | ∗ |s is the extension of the normalization from Qs on Us , consequently, sx ∈ Us is defined for each x ∈ Cs . A quasi-invariant measure µ on X is called pseudo-differentiable for b ∈ Cs , if there exists PD(b, g(x)) for g(x) := µ(−xz + S) for each S ∈ Bco(X) kSkµ < ∞ and each z ∈ Jµb , where Jµb is a K-linear subspace dense in X. For a fixed z ∈ X such measure is called pseudo-differentiable along z. For a one-parameter subfamily of operators B(K, 0, 1) ∋ t 7→ Ut : X → X quasi-invariant measure µ is called pseudo-differentiable for b ∈ Cs , if for each S the same as above there exists PDc (b, g(t)) for a function g(t) := µ(Ut−1 (S), where X may be also a topological group G with a measure quasi-invariant relative to a dense subgroup G′ (see [Lud99t, Lud98s, Lud00a]). 4.2. Let µ, X, and ρ be the same as in Theorem 3.15 and F be a non-Archimedean Fourier transform defined in [VVZ94, Roo78]. Theorems. (1) g(t) := ρ(z +tw, x) j(t) ∈ L(K, v, Ks ) =: V for µ and the Haar measure v with values in Ks , where z and w ∈ Jµ , t ∈ K, j(t) is the characteristic function of a compact subset W ⊂ K. In general, may be k(t) := ρ(z + tw, x) ∈ / V. (2) Let g(t) = ρ(z + tw, x) j(t) with clopen subsets W in K. Then there are µ, for which there exists PD(b, g(t)) for each b ∈ Cs . If g(t) = ρ(z + tw, x), then there are probability measures µ, for which there exists PD(b, g(t)) for each b ∈ Cs with 0 < Re(b) or b = 0. (3) Let S ∈ A f (X, y), kSkµ < ∞, then for each b ∈ U := {b′ : Re b′ > 0 or b′ = 0} there is a pseudo-differentiable quasi-invariant measure µ . Proof. We consider the following additive compact subgroup GT := {x ∈ X|kx( j)k ≤ pk( j) for each j ∈ N} in X, where T = diag{d( j) ∈ K : |d( j)| = p−k( j) for each j ∈ N} is a compact diagonal operator. Then µ from Theorem 3.15 is quasi-invariant relative to the following additive subgroup ST := GT + H, where H := spanK {e( j) : j ∈ N}. The rest of the proof is analogous to that of § I.4.2. 4.2.1. Theorem. Let X be a Banach space of separable type over a locally compact non-Archimedean field K and J be a dense proper K-linear subspace in X such that the embedding operator T : J ֒→ X is compact and nondegenerate, ker(T ) = {0}, b ∈ C. Then a set P b (X, J) of probability Ks -valued measures µ on Bco(X) quasi-invariant and pseudodifferentiable of order b relative to J is of cardinality card(Ks )c . If J ′ , J ′ ⊂ J, is also a dense K-linear subspace in X, then P b (X, J ′ ) ⊃ P b (X, J). Proof. As in § I.3.20.1 choose for T an orthonormal base in X in which T is diagonal and X is isomorphic with c0 over K such that in its standard base {e j : j ∈ N} the operator T is characterized by Te j = a j e j , 0 6= a j ∈ K for each j ∈ N, lim j→∞ a j = 0 (see Appendix A). Take gn from § 3.15.1, where gn ∈ L(K, B f (K), w′ (dx/an ), Ks ), satisfy conditions there and such that there exists limm→∞ PD(b, ∏m n=1 gn (xz)) ∈ L(X, Bco(X), ν, F) by the variable x for each z ∈ J, where x ∈ K, Ks ∪ Cs ⊂ F, F is a non-Archimedean field. Evidently, P b (X, J) ⊂ M (X, J). The family of such sequences of functions {gn : n ∈ N} has the cardinality card(Ks )c , since in L(ν) the subspace of step functions is dense and

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123

the condition of pseudo-differentiability is the integral convergence condition (see § § 4.1 and 4.2). 4.3. Let X be a Banach space over K, b0 ∈ R or b0 = +∞ and suppose that the following conditions are satisfied: (1) T : X → X is a compact operator with ker(T ) = {0}; (2) a mapping F˜ from B(K, 0, 1) to CT (X) := {U : U ∈ C1 (X, X) and (U ′ (x) − I) is a compact operator for each x ∈ X, there is U −1 satisfying the same conditions as U} is given; ˜ = Ut (x) and Φ1Ut (x + h, x) are continuous by t, that is, F˜ ∈ C1 (B(K, 0, 1), (3) F(t) CT (X)); (4) there is c > 0 such that kUt (x) −Us (x)k ≤ kT xk for each x ∈ X and |t − s| < c; (5) for each R > 0 there is a finite-dimensional over K subspace H ⊂ X and c′ > 0 such that kUt (x) −Us (x)k ≤ kT xk/R for each x ∈ X ⊖ H and |t − s| < c′ with (3 − 5) satisfying also for Ut−1 . Theorem. On X there are probability quasi-invariant measures µ which are pseudodifferentiable for each b ∈ Cs with R ∋ Re(b) ≤ b0 relative to a family Ut , where µ are with values in Ks . Proof. From Conditions (2,3) it follows that there is c > 0 such that |det(Ut′ (x))| = |det(Us′ (x))| in L(µ) by x ∈ X and all |t − s| < c, where quasi-invariant and pseudodifferentiable measures µ on X relative to ST may be constructed as in the proof of Theorems 3.15 and 4.2. The final part of the proof is analogous to that of § I.4.3. 4.4. Let X be a Banach space over K, µ be a probability quasi-invariant measure µ : Bco(X) → Ks , that is pseudo-differentiable for a given b with Re(b) > 0, Cb (X) be a space of continuous bounded functions f : X → Ks with k f k := supx∈X | f (x)|. Theorem. For each a ∈ Jµ and f ∈ Cb (X) is defined the following integral: (i) l( f ) =

Z

[ K

Z

f (x)[µ(−λa + dx) − µ(dx)]g(λ, 0, b)v(dλ) X

and there exists a measure ν : Bco(X) → Cs with a bounded variation, moreover, for b ∈ R this ν is a mapping from from Bco(X) into Ks , such that (ii) l( f ) =

Z

f (x)ν(dx), X

where v is the Haar measure on K with values in Qs , moreover, ν is independent from f and may be dependent on a ∈ Jµ . We denote ν =: D˜ ba µ. Proof. From Definition 4.1 and the non-Archimedean analog of the Lebesgue convergence theorem it follows that there exists  Z Z ( f (x + λa) − f (x))g(λ, 0, b)µ(dx) v(dλ) = l( f ), lim j→∞ K\B(K,0,p− j )

X

that is (i) exists. Let (iii) l j (V, f ) :=

Z

K\B(K,0,p− j )

Z

V

 f (x)(µ(−λa + dx) − µ(dx))g(λ, 0, b) v(dλ),

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where V ∈ Bco(X). Then due to construction of § 3.15 for each c > 0 there exists a compact Vc ⊂ X with kX \Vc kνλ < c for each |λ| > 0, where νλ (A) :=

Z

[µ(−λ′ a + A) − µ(A)]g(λ′ , 0, b)v(dλ′ ) K\B(K,0,|λ|)

for A ∈ Bco(X). The rest of the proof is analogous to that of § I.4.4. 4.5. Theorem. Let X be a Banach space over K, | ∗ | = modK (∗) with a probability quasi-invariant measure µ : Bco(X) → Ks and it is satisfied Condition 3.16.(i), suppose µ is pseudo-differentiable and (viii) Jb µ ⊂ T ”Jµ , (Ut : t ∈ B(K, 0, 1)) is a one-parameter family of operators such that Conditions 3.20(i − vii) are satisfied with the substitution of Jµ onto Jµb uniformly by t ∈ B(K, 0, 1), Jµ ⊃ T ′ X, where T ′ , T ” : X → X are compact operators, ker(T ′ ) = ker(T ”) = 0. Moreover, suppose that there are sequences (ix) [k(i, j)] and [k′ (i, j)] with i, j ∈ N, limi+ j→∞ k(i, j) = limi+ j→∞ k′ (i, j) = −∞ and n ∈ ′ N such that |T ”i, j −δi, j | < |Ti,′ j −δi, j |pk(i, j) , |Ui, j −δi, j | < |T ”i, j −δi, j |pk (i, j) and |(U −1 )i, j − ′ δi, j | < |T ”i, j − δi, j |pk (i, j) for each i + j > n, where Ui, j = e˜iU(e j ), (e j : j) is orthonormal basis in X. Then for each f ∈ Cb (X) is defined  Z Z −1 f (x)[µ(Ut (dx)) − µ(dx)] g(t, 0, b)v(dt) (α) l( f ) = B(K,0,1)

X

and there exists a measure ν : Bco(X) → Cs with a bounded total variation [particularly, for b ∈ R it is such that ν : Bco(X) → Ks ] and (β) l( f ) =

Z

f (x)ν(dx), X

where ν is independent from f and may be dependent on (Ut : t), ν =: D˜ Ub ∗ µ. Proof. From the proof of Theorem 3.20 it follows that there exists a se(q) quence Ut of polygonal operators converging uniformly by t ∈ B(K, 0, 1) to Ut and equicontinuously by indices of matrix elements in L(µ). Then there exists R R limq→∞ lim j→∞ B(K,0,1)\B(K,0,p− j [ X f (Ut−1 (x)) − f (x)]g(t, 0, b)µ(dx)]v(dt) for each f ∈ Cb (X).R From conditions (viii, ix), the Fubini and Lebesgue theorems it follows that for νλ := B(K,0,1)\B(K,0,|λ|) [µ(Ut−1 (A)) − µ(A)]g(t, 0, b)v(dt) for A ∈ Bco(X) for each c > 0 there exists a compact Vc ⊂ X and δ > 0 such that kX \ Vc k < c. Indeed, Vc and δ > 0 may be chosen due to pseudo-differentiability of µ, § § 2.30, 3.18, Formula (i), 3.16.(i) and due to continuity and boundednessy (on B(K, 0, 1) ∋ t) of |det Ut′ (Ut−1 )(x))|K satisfying the following conditions Ut−1 (Vc ) ⊂ Vc and k(X \ Vc ) △ (Ut−1 (X \ Vc ))kµ = 0 for each |t| < δ, since Vc = Y ( j) ∩Vc are compact for every j. Repeating proofs 3.20 and 4.4 with the use of Lemma I.2.5 for the family (Ut : t) we get formulas (α, β).

2.5.

Convergence of Ks -Valued Measures

5.1. Definitions, notes and notations. Let S be a normal topological group with the small inductive dimension ind(S) = 0, S′ be a dense subgroup, suppose their topologies are τ and τ′ correspondingly, τ′ ⊃ τ|S′ . Let G be an additive Hausdorff left-R-module, where R is

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a topological ring, R ⊃ Bco(S) be a a ring R ⊃ Bco(S) for Ks -valued measures, M(R,G) be a family of measures with values in G, L(R,G, R) be a family of quasi-invariant mea−1 sure µ : R → G with ρµ (g, x) × µ(dx) := µg (dx) =: µ(gdx), R × G → G be a continuous left action of R on G such that ρµ (gh, x) = ρµ (g, hx)ρµ (h, x) for each g, h ∈ S′ and x ∈ S. Particularly, 1 = ρµ (g, g−1 x)ρµ (g−1 , x), that is, ρµ (g, x) ∈ Ro , where Ro is a multiplicative subgroup of R. Moreover, zy ∈ L for z ∈ R0 with ρzµ (g, x) = zρµ (g, x)z−1 and z 6= 0. We suppose that topological characters and weights S and S′ are countable and each open W in S′ is pre-compact in S. Let P” be a family of pseudo-metrics in G generating the initial uniformity such that for each c > 0 and d ∈ P” and {Un ∈ R : n ∈ N} with ∩{Un : n ∈ N} = {x} there is m ∈ N such that d(µg (Un ), ρµ (g, x)µ(Un )) < cd(µ(Un ), 0) for each n > m, in addition, a limit ρ is independent µ-a.e. on the choice of {Un : n} for each x ∈ S and g ∈ S′ . Consider a sub-ring R′ ⊂ R, R′ ⊃ Bco(S) such that ∪{An : n = 1, . . . , N} ∈ R′ for An ∈ R′ with N ∈ N and S′ R′ = R′ . Then L(R, G, R; R′ ) := {(µ, ρµ (∗, ∗)) ∈ L(R, G, R) : µ − R′ − is regular and for each s ∈ S there are An ∈ R′ , n ∈ N with s = ∩(An : n), {s} ∈ R′ }. For pseudo-differentiable measures µ let S” ⊂ S′ , S” be a dense subgroup in S, τ′ |S” is not stronger than τ” on S” and there exists a neighborhood τ” ∋ W ” ∋ e in which are dense elements lying on one-parameter subgroups (Ut : t ∈ B(K, 0, 1))). We suppose that µ is induced from the Banach space X over K due to a local homeomorphism of neighborhoods of e in S and 0 in X as for the case of groups of diffeomorphisms [Lud96] such that is accomplished Theorem 4.5 for each U∗ ⊂ S” inducing the corresponding transformations on X. In the following case S = X we consider S′ = Jµ and S” = Jµb with (b)

Re(b) > 0 such that Mµ ⊃ Jµ ⊂ (Tµ X)∼ , Jµb ⊂ (Tµ X)∼ with compact operators Tµ and (b)

(b)

Tµ , ker(Tµ ) = ker(Tµ ) = 0 and norms induced by the Minkowski functional PE for (b) E = Tµ B(X, 0, 1) and E = Tµ B(X, 0, 1) respectively. We suppose further that for pseudodifferentiable measures G is equal to Cs ∨ Ks . We denote P(R, G, R,U∗ ; R′ ) := [(µ, ρµ , ηµ ) : (µ, ρµ ) ∈ L(R, G, R; R′ ), µ is pseudo-differentiable and ηµ (t,U∗ , A) ∈ L(K, v, Cs )], where ηµ (t,U∗ , A) = j(t)g(t, 0, b)[µh (Ut−1 (A) − µh (A)], j(t) = 1 for each t ∈ K for S = X; j(t) = 1 for t ∈ B(K, 0, 1), j(t) = 0 for |t|K > 1 for a topological group S that is not a Banach space X over K, v is the Haar measure on K with values in Qs , (Ut : t ∈ B(K, 0, 1)) is an arbitrary one-parameter subgroup. On these spaces L (or P) the additional conditions are imposed: (a) for each neighborhood (implying that it is open) U ∋ 0 ∈ G there exists a neighborhood S ⊃ V ∋ e and a compact subset VU , e ∈ VU ⊂ V , with µ(B) ∈ U (or in addition D˜ Ub ∗ µ(B) ∈ U) for each B, R ∋ B ∈ Bco(S \VU ); (b) for a given U and a neighborhood R ⊃ D ∋ 0 there exists a neighborhood W , S′ ⊃ W ∋ e, (pseudo)metric d ∈ P” and c > 0 such that ρµ (g, x) − ρµ (h, x′ ) ∈ D (or D˜ Ub ∗ (µg − µh )(A) ∈ U for A ∈ Bco(VU ) in addition for P) whilst g, h ∈ W , x, x′ ∈ VU , d(x, x′ ) < c, where (a,b) is satisfied for all (µ, ρµ ) ∈ L (or (µ, ρµ , ηµ ) ∈ P) equicontinuously in (a) on V ∋ Ut ,Ut−1 and in (b) on W and on each VU for ρµ (g, x) − ρµ (h, x′ ) and D˜ Ub ∗ (µg − µh )(A). These conditions are justified, since due to Theorems 3.15, 3.19, 4.3 and 4.5 there exists a subspace Z” dense in Z ′ such that for each ε > 0 and each ∞ > R > 0 there are r > 0 and δ > 0 with |ρν (g, x) − ρν (h, y)| < ε for each kg − hkZ” + kx − ykZ < δ, g, h ∈ B(Z”, 0, r), x, y ∈ B(Z, 0, R), where Z” is the Banach space over K. For a group of diffeomorphisms of a non-Archimedean Banach manifold we have an analogous continuity of ρµ for a subgroup G” of the entire group G (see [Lud96, Lud99t, Lud00a, Lud02b]). By Mo we denote a

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subspace in M, satisfying (a). Henceforth, we imply that R′ contains all closed subsets from S belonging to R, where G and R are complete. For µ : Bco(S) → G by L(S, µ, G) we denote the completion of a space of continuous R f : S → G such that k f kd := suph∈Cb (S,G) d( S f (x)h(x) µ(dx), 0) < ∞ for each d ∈ P”, where Cb (S, G) is a space of continuous bounded functions h : S → G. We suppose that for each sequence ( fn : n) ⊂ L(S, µ, G) for which g ∈ L(S, µ, G) exists with d( fn (x), 0) ≤ d(g(x), 0) for every d ∈ P”, x and n, that fn converges uniformly on each compact subset V ⊂ S with kV kµR > 0 and the following is satisfied: f ∈ L(S, µ, G), limn k fn − f kd = 0 for each d ∈ P” R and S f (x)µ(dx) = limn S fn (x)µ(dx). In the case G = Ks it coincides with L(S, µ, Ks ), hence this supposition is the Lebesgue theorem. By Y (v) we denote L(K, v, Cs ). Now we may define topologies and uniformities with the help of corresponding bases ′ ′ ′ (see below) on L ⊂ GR × RSo ×S =: Y (or P ⊂ GR × RSo ×S × GS ×K×R =: Y , Ro ⊂ R \ {0}. There are the natural projections π : L (∨ P) → Mo , π(µ, ρµ (∗, ∗) (∨ ,ηµ )) = µ, ξ : L (∨ ′ ′ P) → RS ×S , ξ(µ, ρµ , (∨ ηµ )) = ρµ , ζ : P → GS ×K×R , ζ(µ, ρµ , ηµ ) = ηµ . Let H be a filter on L or P, U = U ′ ×U” or U = U ′ ×U” ×U”′ , U ′ and U” be elements of uniformities on G, R and Y (v) correspondingly, τ′ ∋ W ∋ e, τ ∋ V ⊃ VU ′ ∋ e, VU ′ is compact. By [µ] we denote (µ, ρµ ) for L or (µ, ρµ , ηµ ) for P, Ω := L ∨ P, [µ](A,W,V ) := [µg (A), ρµ (g, x), ∨ ηµg (t,U∗ , A)| g ∈ W, x ∈ V, ∨ t ∈ K]. We consider A ⊂ R, then W(A,W,VU ′ ;U) := {([µ], [ν]) ∈ Ω2 |([µ], [ν])(A,W,VU ′ ) ⊂ U};

(1)

W(S;U) := {([µ], [ν]) ∈ Ω2 |{(B, g, x) : ([µ], [ν])(B, g, x)) ∈ U} ∈ S},

(2)

where S is a filter on R × S′ × Sc , Sc is a family of compact subsets V ′ ∋ e. W(F,W,V ;U) := {([µ], [ν]) ∈ Ω2 |{B : ([µ], [ν])(B, g, x) ∈U, g ∈W, x ∈V } ∈ F},

(3)

where F is a filter on R (see also § 2.1 and 4.1[Con84]); W(A, G;U) := {([µ], [ν]) ∈ Ω2 | {(g, x) : ([µ], [ν])(B, g, x) ∈ U, B ∈ A} ∈ G},

(4)

where G is a filter on S′ × Sc ; suppose U ⊂ R × τ′e × Sc , Φ is a family of filters on R × S′ × Sc or R × S′ × Sc ×Y (v) (generated by products of filters ΦR × ΦS′ × ΦSc on the corresponding spaces), U′ be a uniformity on (G, R) or (G, R,Y (v)), F ⊂ Y . A family of finite intersections of sets W(A,U) ∩ (F × F) (see (1)), where (A,U) ∈ U × U′ (or W(F,U) ∩ (F × F) (see (2)), where (F,U) ∈ (Φ × U′ ) generate by the definition a base of uniformity of U-convergence (Φ-convergence respectively) on F and generate the corresponding topologies. For these uniformities are used notations (i) FU and FΦ ;

FR×W ×V is for F with the uniformity of uniform convergence

on R ×W ×V , where W ∈ τ′e , V ∈ Sc , analogously for the entire space Y ; (ii) FA denotes the uniformity (or topology) of pointwise convergence for A ⊂ R × τ′e × Sc =: Z, for A = Z we omit the index (see formula (1)). Henceforward, we ˜ use H′ instead of H in 4.1.24[Con84], that is, H′ (A, R)-filter on R generated by the base ′ ′ ′ ˜ ˜ ˜ [(L ∈ R : L ⊂ A \ K ) : K ∈ R, K ⊂ A], where R ⊂ R and R is closed relative to the finite unions.

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For example, let S be a locally K-convex space, S′ be a dense subspace, G be a locally L-convex space, where K, L are fields, R = B(G) be a space of bounded linear operators on G, Ro = GL(G) be a multiplicative group of invertible linear operators. Then others possibilities are: S = X be a Banach space over K, S′ = Jµ , S” = Jµb as above; S = G(t), S′ ⊃ S” are dense subgroups, G = R be the field Ks ( s 6= p), M be an analytic Banach manifold over K ⊃ Qp (see [Lud96]). The rest of the necessary standard definitions are recalled further when they are used. 5.2. Lemma. Let R be a quasi-δ-ring with the weakest uniformity in which ˆ C (R, S′ × Sc ). Then L(R, G, R, R′ )Φ (or each µ ∈ M is uniformly continuous and Φ ⊂ Φ ′ P(R, G, R,U∗ ; R )Φ )) is a topological space on which Ro acts continuously from the right. Proof. It is analogous to that of § I.5.2 using Definition 4.1 for pseudo-differentiable f . ˆ 4 -filter on Mo (R,G; R′ ), {An } be a disjoint Θ(R)5.3. Proposition. (1). Let T be a Φ sequence, Σ be the elementary filter on R generated by {An : n ∈ N} and φ : Mo × R → G with φ(µ, A) = µ(A). Then φ(T × Σ) converges to 0. (2). Moreover, let U be a base of neighborhoods of e ∈ S′ , φ : L → G × R, φ(µ, A, g) := (µg (A), ρµ (g, x)), where x ∈ A. Then (0, 1) ∈ lim φ(T × Σ × U). (3). If T is a Φ4 -filter on P(R, G, R,U∗ ; R′ ), ψ(µ, B, g,t,U∗ ) = [µ(B); ρµ (g, x); ηµg (t,U∗ , B)], then (0, 1, 0) ∈ lim ψ(T × Σ × U) for each given U∗ ∈ S”, where Σ and U as in (1, 2). Proof. The proof is analogous to that of § I.5.3 with the use of the Lebesgue convergence theorem. ˆ 4 -filter on L (or P) with the topology F (see 5.1(ii)), 5.4. Proposition. Let H be a Φ ′ ′ A ∈ R, τG ∋ U ∋ 0, H (A, R ) ∈ Ψ f (R). Then there are L ∈ H, K˜ ∈ R′ and an element ˜ ⊂ A, L = [(µ, ρµ (g, x)) : M := πMo (L) ∋ of the uniformity U for LR′ or PR′ such that K µ, πτ′e (L) =: W ∋ g (or (µ, ρµ , ηµ (∗, ∗,U∗ )) and additionally D˜ Ub ∗ µ = PD(b, ηµ ))], e ∈ W ∈ τ′ , µg (B)−νh (C) ∈ U (or in addition (D˜ Ub ∗ µg (B))−(D˜ Ub ∗ νh (C)) ∈ U) for K˜ ⊂ B ⊂ A, K˜ ⊂ C ⊂ A for each ([µ], [ν]) ∈ L¯ 2 ∩ U, where L¯ := cl(L, LR′ ) (or cl(L, PR′ )), πMo is a projector from L into Mo . Proof. Repeating the proof of § I.5.4 we get µg (B) − µ(B) ∈ U ′ , νh (C) − ν(C) ∈ U ′ and for 3U ′ ⊂ U we get 5.4 for L. From Theorems 4.4 and 4.5, §5.1, the Egorov conditions and the Lebesgue theorem we get 5.4 for P, since µ are probability measures and LR′ (or PR′ ) correspond to uniformity from § 5.1(ii) with A = R′ × τ′e × Sc . Indeed, µg (A) − νh (A) = R (µg (A) − µg (VU ′ )) + (µg (VU ′ ) − νh (VU ′ )) + (νh (VU ′ ) − νh (A)), µg (A) = A ρµ (g, x)µ(dx) for each A ∈ Bco(S), for each τG ∋ U ′ ∋ 0 there exists a compact subset VU′ ⊂ A with µg (B) ∈ U ′ for each B ∈ B f (A \VU ′ ) ∩ Bco(S) and the same for νh (due to the condition in § 5.1 that R′ contains Bco(S)). At first we can consider A ∈ Bco(S), then use R′ -regularity of measures and σR′ ⊃ Bco(S). From the separability of S, S′ and the equality of their topological weights to ℵ0 , restrictions 5.1.(a,b) it follows that there exists a sequence of partitions Zn = [(xm , Am ) : S m, xm ∈ Am ] for each A ∈ Bco(S), Ai ∩ A j = 0/ for each i 6= j, m Am = A, Am ∈ Bco(S), such that limn→∞ (µg (A) − ∑ j ρµ (g, x j )µ(A j )) = 0 and the same for ν, moreover, for VU ′ each Zn may be chosen finite. Then there exists W ∈ τ′e with W × (S \V 2 ) ⊂ (S \V ), τe ∋ V ⊂ V 2 , νg (B) and µg (B) ∈ U ′ for each B ∈ B f (S \ V 2 ) ∩ Bco(S) (for G = Ks respectively) and

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g ∈ W (see 5.1.(a)). Then from A = [A ∩ (S \ V 2 )] ∪ [A ∩ V 2 ] and the existence of compact VU′ ′ ⊂ V with µ(E) ∈ U ′ for each E ∈ B f (V \VU′ ′ ) ∩ Bco(S) and the same for ν, moreover, (VU′ ′ )2 is also compact, it follows that µg (B) − νh (C) ∈ U for 9U ′ ⊂ U, since R′ ⊃ Bco(S), where W satisfies the following condition µg (VU′ ′ ) − νh (VU′ ′ ) ∈ U ′ for VU′ ′ ⊂ V 2 due to § 5.1.(b), µ(B) − ν(C) ∈ U ′ , WVU′ ′ ⊂ (VU′ ′ )2 due to pre-compactness of W in S. Since pseudodifferentiable measures are also quasi-invariant, hence for them 5.4 is true. Now let [µ] ∈ lim H,R A ∈ Bco(S), then ηµ ∈ lim ζ(H) in Y (v) and there exists a sequence ηµn such that K ηµn (λ,U∗ , A)v(dλ) = D˜ Ub ∗ µn (A) due to § § 4.4 or 5.1 and R limn→∞ D˜ Ub ∗ µn (A) = K ηµ (λ,U∗ , A)v(dλ) =: κ(A) due to the Lebesgue theorem. From / B ∈ Bco(S) it follows that ν(A) ηµ (λ,U∗ , A ∪ B) = η(λ,U∗ , A) + η(λ,U∗ , B) for A ∩ B = 0, R is the measure on Bco(S), moreover, κ(A) = D˜ Ub ∗ µ(A). Since µg (A) = A ρµ (g, x)µ(dx) for A ∈ Bco(S) for g ∈ S′ , then ηµg (λ,U∗ , A) = j(λ)g(λ, 0, b)[µg (A) − µg (Uλ−1 A)] = j(λ)g(λ, 0, b)

Z

A

ρµ (g, x)[µ(dx) − µUλ (dx)]

and in view of the Fubini theorem there exists  Z Z Uλ b g ˜ ρµ (g, x) j(λ)g(λ, 0, b)[µ(dx) − µ (dx)] v(dλ) DU∗ µ (A) = A

K

, where j(t) = 1 for S = X and j(t) is the characteristic function of B(K, 0, 1) for S that is not the Banach space X. Then µ-a.e. D˜ Ub ∗ µg (dx)/D˜ Ub ∗ µ(dx) coincides with ρµ (g, x) due to ˆ 4 -filter in P. Then 5.1(a, b), hence, (D˜ Ub ∗ µg , ρµg ) generate the Φ4 -filter in L arising from the Φ b g h g h ˜ we estimate DU∗ (µ − ν )(A) as above µ (A) − ν (A). Therefore, we find for the Φ4 -filter corresponding L, since there exists δ > 0 such that Uλ ∈ W for each |λ| < δ. For Φ4 -filter we use the corresponding finite intersections W1 ∩ · · · ∩ Wn = W , where W j correspond to the Φ4 -filters H j . Note. The formulations and proofs of § § 5.5-5.10 (see Chapter I) are quite analogous for real-valued and Ks -valued measures due to preceding results.

2.6.

Measures with Particular Properties

1. Theorem. Let X be a complete separable ultra-uniform space and let K be a locally compact field. Then for each marked b ∈ Cs there exists a nontrivial Fvalued measure µ on X which is a restriction of a measure ν in a measure space (Y, Bco(Y ), ν) = lim{(Ym , Bco(Ym ), νm ), f¯nm , E} on X and each νm is quasi-invariant and pseudo-differentiable for b ∈ Cs relative to a dense subspace Y ′ m , where Yn := c0 (K, αn ), f¯nm : Ym → Yn is a normal (that is, K-simplicial non-expanding) mapping for each m ≥ n ∈ E, f¯nm |Pm = fnm . Moreover, if X is not locally compact, then the family F of all such µ contains a subfamily G of pairwise orthogonal measures with the cardinality card(G ) = card(F)c , c := card(Qp ). Proof. Choose a polyhedral expansion of X in accordance with Theorem B.2.18. Let either Qp ⊂ K for char(K) = 0 or Fp (θ) ⊂ K for char(K) = p, s 6= p are prime numbers, Qs ⊂ F, where F is a non-Archimedean field complete relative to its uniformity. On each

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Xn take a probability F-valued measure νn such that kXn \ Pn kνn < εn , ∑n∈E εn < 1/5. In accordance with § 3.5.1 and § 4.2.1 each νn can be chosen quasi-invariant and pseudodifferentiable for b ∈ Cs relative to a dense K-linear subspace Y ′ n , since each normal mapping fnm has a normal extension on Ym supplied with the uniform polyhedra structure. Since E is countable and ordered, then a family νn can be chosen by transfinite induction consistent, that is, f¯nm (νm ) = νn for each m ≥ n in E, f¯nm (Y ′ m ) = Y ′ n . Then X = lim{Pm , fnm , E} ֒→ Y . Since f¯nm are K-linear, then ( f¯nm )−1 (Bco(Yn )) ⊂ Bco(Ym ) for each S m ≥ n ∈ E. Therefore, ν is correctly defined on the algebra n∈E fn−1 (Bco(Yn )) of subsets of Y , where fn : X → Xn are K-linear continuous epimorphisms. Since ν is nontrivial and kνk is bounded by 1, then by the non-Archimedean analog of the Kolmogorov theorem 2.39 ν has an extension on the algebra Bco(Y ) and hence on its completion A f (Y, ν). Put Y ′ := lim{Y ′ m , f¯nm , E}. Then νm on Ym is quasi-invariant and pseudo-differentiable for b ∈ Cs relative to Y ′ m . From ∑n εn < 1/5 it follows, that 1 ≥ kXkµ ≥ ∏n (1 − εn ) > 1/2, hence µ is nontrivial. To prove the latter statement use the non-Archimedean analog of the Kakutani theorem for ∏n Yn and then consider the embeddings X ֒→ Y ֒→ ∏n Yn such that projection and subsequent restriction of the measure ∏n νn on Y and X are nontrivial, which is possible due to the proof given above. If ∏n νn and ∏n ν′ n are orthogonal on ∏n Yn , then they give ν and ν′ orthogonal on X. 2. Definitions and Notes. Let spaces X and Y be as in § 1.6.4. Consider a nonArchimedean field F such that Ks ⊂ F and with the normalization group ΓF = (0, ∞) ⊂ R and F is complete relative to its uniformity (see [Dia84, Esc95]). Then a measure µ = µq,B,γ on X with values in Ks is called a q-Gaussian measure, if its characteristic functional µˆ with values in F has the form s s µˆ (z) = s[B(vq (z),vq (z))] χγ (z) on a dense K-linear subspace Dq,B,X in X ∗ of all continuous K-linear functionals z : X → K of the form z(x) = z j (φ j (x)) for each x ∈ X with vsq (z) ∈ DB,Y , where B is a nonnegative definite bilinear R-valued symmetric functional on a dense R-linear subspace DB,Y in Y ∗ , B : D2B,Y → R, j ∈ ϒ may depend on z, z j : X j → K is a continuous K-linear functional such that z j = ∑k∈α j ekj zk, j is a countable convergent series such that zk, j ∈ K, ekj is a continuous K-linear functional on X j such that ekj (el, j ) = δkl is the Kroneker delta symbol, el, j is the standard orthonormal (in the non-Archimedean sense) basis in c0 (α j , K), vsq (z) = vsq (z j ) := {|sq ord p (zk, j )/2 |s : k ∈ α j }. It is supposed that z is such that vsq (z) ∈ l2 (α j , R), where q is a positive constant, χγ (z) : X → Ts is a continuous character such that χγ (z) = χ(z(γ)), γ ∈ X, χ : K → Ts is a nontrivial character of K as an additive group (see Chapter 9 in [Roo78] and § 2.5 above). 3. Proposition. A q-Gaussian quasi-measure on an algebra of cylindrical subsets S −1 π j j (R j ), where X j are finite-dimensional over K subspaces in X, is a measure on a covering ring R of subsets of X (see § 2.36). Moreover, a correlation operator B is of class L1 , that is, Tr(B) < ∞, if and only if each finite dimensional over K projection of µ is a q-Gaussian measure (see §2.1). Proof. From Definition 2 it follows, that each one dimensional over K projection µxK of a measure µ satisfies Conditions 2.1.(i − iii) the covering ring Bco(K), where 0 6= x = ek,l ∈ Xl . Therefore, µ is defined and finite additive on a cylindrical algebra

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l −1 U := k1 ,...,kn ;l φ−1 l [(φk1 ,...,kn ) (Bco(spanK {ek1 ,l , . . . , ekn ,l }))], where φlk1 ,...,kn : Xl → spanK (ek1 ,l , . . . , ekn ,l ) is a projection. This means that µ is a bounded quasi-measure on U. Since µˆ (0) = 1, then µ(X) = 1. The characteristic functional µˆ satisfies Conditions 2.5.(3, 5). In view of the non-Archimedean analog of the Bochner-Kolmogorov theorem (see § 2.21 above) and Theorem 2.37 µ has an extension to a probability measure on a covering ring R of subsets of X containing U. Suppose that B is of class L1 . Then B(vq (z), vq (z)) and hence µˆ (z) is correctly defined for each z ∈ Dq,B,X . The set Dq,B,X of functionals z on X from § 2 separates points of X. From Definition 2 it follows, that µˆ (y) is continuous. Consider a diagonal compact operator T in the standard orthonormal base, Tek,l = ak,l ek,l , limk+l→∞ ak,l = 0. Since B is continuous, then the corresponding to B correlation operator E is a bounded K-linear operator on Y , kEk < ∞. For each ε > 0 there exist δ > 0 and T such that max(1, kEk)δ < ε and |ak,l | < δ for each k + l > N, where N is a marked natural number, therefore, kE|spanK {ek,l :k+l>N} k < ε. Hence for each ε > 0 there exists a compact operator T such that from |˜zT z| < 1 it follows, |ˆµ(y) − µˆ (x)| < ε for each x − y = z, where x, y, z ∈ Y ∗ . Therefore, by Theorem 2.30 the characteristic functional µˆ defines a probability Radon measure on Bco(X). Vice versa suppose that each finite dimensional over K projection of µ is a measure of the same type. If for a given one dimensional over K subspace W in X it is the equality B(vq (z), vq (z)) = 0 for each z ∈ W , then the projection µW of µ is the atomic measure with one atom. Show B ∈ L1 (c0 (ω0 , K)) and γ ∈ c0 (ω0 , K). Let 0 6= x ∈ X and consider the projection πx : X → xK. Since µxK is the measure on Bco(xK), then its characteristic functional satisfies Conditions of Theorem 2.30. Then µˆ for xK gives the same characteristic functional of the type

S

s

2

µˆ xK (z) = s[bx (vq (z)) ] χδx (z) for each z ∈ xK, where bx > 0 and δx ∈ K are constants depending on the parameter 0 6= x ∈ X. Since x and z are arbitrary, then this implies, that B ∈ L1 and γ ∈ c0 (ω0 , K). 4. Corollary. A q-Gaussian measure µ from Proposition 3 with Tr(B) < ∞ is quasiinvariant and pseudo-differentiable for some b ∈ Cs relative to a dense subspace Jµ ⊂ Mµ = {x ∈ X : vsq (x) ∈ E 1/2 (Y )}. Moreover, if B is diagonal, then each one-dimensional projection µg has the following characteristic functional: (i) µˆ g (h) = s(∑ j β j |g j |

q )|h|q

χg(γ) (h),

where g = (g j : j ∈ ω0 ) ∈ c0 (ω0 , K)∗ , β j > 0 for each j. Proof. Using the projective limit reduce consideration to the Banach space X. Take a prime number s such that s 6= p and consider a field Ks such that K is compatible with Ks , which is possible, since K is a finite algebraic extension of Qp and it is possible to take in particular Ks = Qs . Recall that a group G for which o(G) ⊂ o(TK ) is called compatible with K, where o(G) denotes the set of all natural numbers for which G has an open subgroup U such that at least one of the elements of the quotient group G/U has order n, T denotes the group of all roots of 1 and TK denotes its subgroup of all elements whose orders are not divisible by the characteristic p of the residue class field k of K. A character of G is a continuous homomorphism f : G → T. Under point-wise multiplication characters form

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a group denoted by Gˆ. . A group G is called torsional, if each compact subset V of G is contained in a compact subgroup of G. Theorem 9.14 [Roo78] states: if the field K is locally compact, then K is torsional, K and Kˆ. are isomorphic as topological groups. Taking any non-constant character φ of K and setting φs (t) := φ(st) with s,t ∈ K one gets an isomorphism s 7→ φs and a homeomorphism of K onto Kˆ. . In view of this theorem Kˆ. is isomorphic with K. A K-valued character of a group G is a continuous homomorphism f : G → TK . The family of all K-valued characters form a group denoted by Gˆ.K . Since K is compatible with Ks and limn→∞ pn = 0, then Kˆ. is isomorphic with Kˆ.Ks . If G is a locally compact torsional group compatible with K, then the Fourier-Stieltjes transform of a tight measure µ ∈ M(G) is the mapping µˆ : Gˆ.K → K defined by the formula: R µˆ (g) := G χ(x)µ(dx), where χ ∈ Gˆ.K . Moreover, the Fourier-Stieltjes transform induces a Banach algebra isomorphism L(G, R , w, K) with C∞ (Gˆ.K , K), where w is a nontrivial Haar K-valued measure on G. The just above formulated statement is proved in the Schikhof’s theorem (see also § 9.21 in [Roo78]). Therefore, in this situation there exists the Banach algebra isomorphism of L(K, R , w, Ks ) with C∞ (Kˆ.Ks , Ks ). Therefore, from the proof above and Theorem 3.5 it follows, that the measure µq,B,γ is quasi-invariant relative to shifts on vectors from the dense subspace X ′ in X such that X ′ = {x ∈ X : vsq (x) ∈ E 1/2 (Y )}, which is K-linear, since B is R-bilinear and B(y, z) =: (Ey, z) for each y, z ∈ Y and vsq (ax) = |a|q/2 vsq (x) and vsq (x j + t j ) ≤ max(vsq (x j ), vsq (t j )) for each x,t ∈ X and each a ∈ K, where E is nondegenerate positive definite of trace class R-linear operator on Y , x = ∑ j x j e j , x j ∈ K, since l2∗ = l2 and E can be extended from DB,Y on Y . Consider sa+ib as in § 4.1. Mention, that |(|z| p )|s = 1 for each z ∈ K, where the field K is compatible with Ks . The pseudo-differential operator has the form: PD(b, f (x)) :=

Z

[ f (x) − f (y)]s(−1−b)×ord p (x−y) w(dy), K

where w is the Haar Ks -valued measure on Bco(K), b ∈ Cs , particularly, also for f (x) := µ(−xz + A) for a given z ∈ X ′ , A ∈ Bco(X), where x, y ∈ K. Using the Fourier-Stieltjes transform write it in the form: PD(b, f (x)) = Fv−1 (ξ(v)ψ(v)), where ξ(v) := [Fy ( f (x) − f (y))](v), ψ(v) := [Fy (s(−1−b)×ord p (y) )](v), Fy means the Fourier-Stieltjes operator by the variable y. Denoting A−xz =: SRwe can consider f (x) = 0 and Rf (y) = µ((x−y)z+S)−µ(S), since S ∈ Bco(X). Then f (y) = S (µ((x−y)+dg)−µ(dg)) = S [ρµ (y−x, g)−1]µ(dg). The constant function h(g) = 1 is evidently pseudo-differentiable of order b for each b ∈ Cs . Hence the pseudo-differentiability of µ of order b follows from the existence of pseudodifferential of the quasi-invariance factor ρµ (y, g + x) of order b for µ-almost every g ∈ X. In view of Theorem 3.5 and the Fourier-Stieltjes operator isomorphism of Banach algebras L(K, R , w, Ks ) and C∞ (Kˆ.Ks , Ks ) the pseudo-differentiability of ρµ follows from the existence of F −1 (ˆµψ), where µˆ is the characteristic functional of µ. We have (ii) F( f )(y) =

Z

K

χ(xy) f (x)w(dx)

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Sergey V. Ludkovsky =

Z

K

χ(z) f (z/y)[|y| p ]−1 w(dz)

for each y 6= 0, where x, y, z ∈ K, particularly, for f (x) = s−(1+b)×ord p (x) we have f (z/y) = f (z) f (−y) and F( f )(y) = ΓK,s (1 + b) f (−y)|y|−1 p , where (iii) ΓK,s (b) :=

Z

K

χ(z)s−b×ord p (x) w(dz),

f (−y) = s(1+b)×ord p (y) , since ord p (z/y) = ord p (x) − ord p (y). For a nontrivial character of an order m ∈ Z from the definition it follows, that ΓK,s (b) 6= 0 for each b with Re(b) 6= 0, since |s−bn |s = sRe(b)n for each n ∈ Z. Therefore, −(1+Re(b))×ord p (y)) for each y 6= 0, since ψ(y) = s(1+b)×ord p (y) |y|−1 p , consequently, |ψ(y)|s = s s s |(|y| p )|s = 1. On the other hand, |ˆµ(z)| = s−B(vq (z),vq (z)) and F −1 (ˆµψ) exists for each b ∈ Cs with Re(b) > −1, since Tr(B) < ∞, which is correct, since Cs is algebraically isomorphic with C and ΓUs ⊃ (0, ∞). 5. Corollary. Let X be a complete locally K-convex space of separable type over a local field K, then for each constant q > 0 there exists a nondegenerate symmetric positive definite operator B ∈ L1 such that a q-Gaussian quasi-measure is a measure on Bco(X) and each its one dimensional over K projection is absolutely continuous relative to the nonnegative Haar measure on K. Proof. It is analogous to that of Corollary I.6.8. For each K-linear finite dimensional over K subspace S a projection µS of µ on S ⊂ X exists and its density µS (dx)/w(dx) relative to the nondegenerate Ks -valued Haar measure w on S is the inverse Fourier-Stieltjes transform F −1 (ˆµ|S∗ ) of the restriction of µˆ on S∗ . For B ∈ L1 each one dimensional projection of µ corresponding to µˆ has a density that is a continuous function belonging to L(K, Bco(K), w, Ks ). 6. Proposition. Let µq,B,γ and µq,E,δ be two q-Gaussian measures with correlation operators B and E of class L1 , then there exists a convolution of these measures µq,B,γ ∗ µq,E,δ , which is a q-Gaussian measure µq,B+E,γ+δ . Proof. It is analogous to that of I.6.9 with the substitution of B f (X) on Bco(X). 6.1. Remark and Definition. A measurable space (Ω, F) with a probability Ks -valued measure λ on a covering ring F of a set Ω is called a probability space and it is denoted by (Ω, F, λ). Points ω ∈ Ω are called elementary events and values λ(S) probabilities of events S ∈ F. A measurable map ξ : (Ω, F) → (X, B) is called a random variable with values in X, where B is a covering ring such that B ⊂ Bco(X), Bco(X) is the ring of all clopen subsets of a locally K-convex space X, ξ−1 (B) ⊂ F, where K is a non-Archimedean field complete as an ultra-metric space. The random variable ξ induces a normalized measure νξ (A) := λ(ξ−1 (A)) in X and a new probability space (X, B, νξ ). Let T be a set with a covering ring R and a measure η : R → Ks . Consider the following Banach space Lq (T, R , η, H) as the completion of the set of all R -step functions f : T → H relative to the following norm: (1) k f kη,q := supt∈T k f (t)kH Nη (t)1/q for 1 ≤ q < ∞ and (2) k f kη,∞ := sup1≤q 0. For example, T may be a subset of R. Let Rd be the field R supplied with the discrete topology. Since the cardinality card(R) = c = 2ℵ0 , then there are bijective mappings of R on Y1 := {0, . . . , b}N and also on Y2 := NN , where b is a positive integer number. Supply {0, . . . , b} and N with the discrete topologies and Y1 and Y2 with the product topologies. Then zero-dimensional spaces Y1 and Y2 supply with covering separating rings R 1 and R 2 contained in Bco(Y1 ) and Bco(Y2 ) respectively. Certainly such separating covering ring in R induced from Y1 or Y2 is not related with the standard (Euclidean) metric in R. Therefore, for the space Lq (T, R , η, H) we can consider t ∈ T as the real time parameter. If T ⊂ F with a non-Archimedean field F, then we can consider the non-Archimedean time parameter. If T is a zero-dimensional T1 -space, then denote by Cb0 (T, H) the Banach space of all continuous bounded functions f : T → H supplied with the norm: (4) k f kC0 := supt∈T k f (t)kH < ∞. If T is compact, then Cb0 (T, H) is isomorphic with the space C0 (T, H) of all continuous functions f : T → H. For a set T and a complete locally K-convex space H over K consider the product K-convex space H T := ∏t∈T Ht in the product topology, where Ht := H for each t ∈ T . Then take on either X := X(T, H) = Lq (T, R , η, H) or X := X(T, H) = Cb0 (T, H) or on X = X(T, H) = H T a covering ring B such that B ⊂ Bco(X). Consider a random variable ξ : ω 7→ ξ(t, ω) with values in (X, B), where t ∈ T . Events S1 , . . . , Sn are called independent in total if P(∏nk=1 Sk ) = ∏nk=1 P(Sk ). Sub-rings Fk ⊂ F are said to be independent if all collections of events Sk ∈ Fk are independent in total, where k = 1, . . . , n, n ∈ N. To each collection of random variables ξγ on (Ω, F) with γ ∈ ϒ is related the minimal ring Fϒ ⊂ F with respect to which all ξγ are measurable, where ϒ is a set. Collections {ξγ : γ ∈ ϒ j } are called independent if such are Fϒ j , where ϒ j ⊂ ϒ for each j = 1, . . . , n, n ∈ N. Consider T such that card(T ) > n. For X = Cb0 (T, H) or X = H T define X(T, H; (t1 , . . . ,tn ); (z1 , . . . , zn )) as a closed sub-manifold in X of all f : T → H, f ∈ X

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such that f (t1 ) = z1 , . . . , f (tn ) = zn , where t1 , . . . ,tn are pairwise distinct points in T and z1 , . . . , zn are points in H. For X = Lq (T, R , η, H) and pairwise distinct points t1 , . . . ,tn in T with Nη (t1 ) > 0, . . . , Nη (tn ) > 0 define X(T, H; (t1 , . . . ,tn ); (z1 , . . . , zn )) as a closed sub-manifold which is the completion relative to the norm k f kη,q of a family of R -step functions f : T → H such that f (t1 ) = z1 , . . . , f (tn ) = zn . In these cases X(T, H; (t1 , . . . ,tn ); (0, . . . , 0)) is the proper K-linear subspace of X(T, H) such that X(T, H) is isomorphic with X(T, H; (t1 , . . . ,tn ); (0, . . . , 0)) ⊕ H n , since if f ∈ X, then f (t) − f (t1 ) =: g(t) ∈ X(T, H;t1 ; 0) (in the third case we use that T ∈ R and hence there exists the embedding H ֒→ X). For n = 1 and t0 ∈ T and z1 = 0 we denote X0 := X0 (T, H) := X(T, H;t0 ; 0). 6.2. Definitions. We define a (non-Archimedean) stochastic process w(t, ω) with values in H as a random variable such that: (i) the differences w(t4 , ω) − w(t3 , ω) and w(t2 , ω) − w(t1 , ω) are independent for each chosen (t1 ,t2 ) and (t3 ,t4 ) with t1 6= t2 , t3 6= t4 , such that either t1 or t2 is not in the twoelement set {t3 ,t4 }, where ω ∈ Ω; (ii) the random variable ω(t, ω) − ω(u, ω) has a distribution µFt,u , where µ is a probability Ks -valued measure on (X(T, H), B) from § 6.1, µg (A) := µ(g−1 (A)) for g : X → H such that g−1 (R H ) ⊂ B and each A ∈ R H , a continuous linear operator Ft,u : X → H is given by the formula Ft,u (w) := w(t, ω) − w(u, ω) for each w ∈ Lq (Ω, F, λ; X), where 1 ≤ q ≤ ∞, R H −1 is a covering ring of H such that Ft,u (R H ) ⊂ B for each t 6= u in T ; (iii) we also put w(0, ω) = 0, that is, we consider a K-linear subspace Lq (Ω, F, / X0 is the closed subspace of X as in § 6.1. λ; X0 ) of Lq (Ω, F, λ; X), where Ω 6= 0, 7. Definition. Let B and q be as in § 2 and denote by µq,B,γ the corresponding qGaussian Ks -valued measure on H. Let ξ be a stochastic process with a real time t ∈ T ⊂ R (see Definition 6.2), then it is called a non-Archimedean q-Wiener process with real time (and controlled by Ks -valued measure), if (ii)′ the random variable ξ(t, ω)−ξ(u, ω) has a distribution µq,(t−u)B,γ for each t 6= u ∈ T . Let ξ be a stochastic process with a non-Archimedean time t ∈ T ⊂ F, where F is a local field, then ξ is called a non-Archimedean q-Wiener process with F-time (and controlled by Ks -valued measure), if (ii)” the random variable ξ(t, ω) − ξ(u, ω) has a distribution µq,ln[χF (t−u)]B,γ for each t 6= u ∈ T , where χF : F → T is a continuous character of F as the additive group (see §2.5). 8. Proposition. For each given q-Gaussian measure a non-Archimedean q-Wiener process with real (F respectively) time exists. Proof. In view of Proposition 6 for each t > u > b a random variable ξ(t, ω) − ξ(b, ω) has a distribution µq,(t−b)B,γ for real time parameter. If t, u, b are pairwise different points in F, then ξ(t, ω) − ξ(b, ω) has a distribution µq,ln[χF (t−b)]B,γ , since ln[χF (t − u)] + ln[χF (u − (q)

b)] = ln[χF (t − b)]. This induces the Markov quasi-measure µx0 ,τ on (∏t∈T (Ht , Ut )), where Ht = H and Ut = Bco(H) for each t ∈ T . In view of Theorem 2.39 there exists an abstract probability space (Ω, F, λ), consequently, the corresponding space L(Ω, F, λ, Ks ) exists. 9. Proposition. Let ξ be a q-Gaussian process with values in a Banach space H = c0 (α, K) a time parameter t ∈ T (controlled by a Ks -valued measure) and a positive definite correlation operator B of trace class and γ = 0, where card(α) ≤ ℵ0 , either T ⊂ R or T ⊂ F. Then either (i)

lim Mt [vsq (e1 (ξ(t, ω))2 + · · · + vsq (eN (ξ(t, ω)))2 ] = tTr(B) or

N∈α

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135

lim Mt [vsq (e1 (ξ(t, ω))2 + · · · + vsq (eN (ξ(t, ω))2 ] = [ln(χF (t))]Tr(B) respectively.

N∈α

Proof. Define Us -valued moments q

mk (e j1 , . . . , e jk ) :=

Z

H

vs2q (e j1 (x)) · · · vs2q (e jk (x))µq,B,γ (dx)

for linear continuous functionals e j1 , . . . , e jk on H such that el (e j ) = δlj , where {e j : j ∈ α} is the standard orthonormal base in H. Consider the operator ˆ (iii) P ∂u ψ(x) := F −1 ( fˆu−1 (y)ψ(y)|y| p )(x), where fu (x) := s−(1+u)×ord p (x) /ΓK,s (1 + u) and F( fu )(y) = ΓK,s (1 + u) fu (−y)|y|−1 p (see §4), where F denotes the Fourier-Stieltjes operator defined with the help of the Ks -valued Haar ˆ Re(u) 6= −1, ψ : K → Ks . Then measure w on Bco(K), F(ψ) =: ψ, (iv) P ∂u fb (x) = F −1 (ΓK,s (u) fu−1 (−y)ΓK,s (1 + b) fb (−y)|y|−1 p ) = f (u+b) (x) for each u with Re(u) 6= 0, since K,s (1 + u + b))−1 s−(1+u+b)×ord p (−y) (x). F −1 (s−(1+u+b)×ord p (−y) |y|−1 p )(x) = (Γ For u = 1 we write shortly P ∂1 = P ∂ and P ∂uj means the operator of partial pseudodifferential (with weight multiplier) given by Equation (iii) by the variable x j . A function ψ for which P ∂uj ψ exists is called pseudo-differentiable (with weight multiplier) of order u by variable x j . Then q/2

m2k (e j1 , . . . , e j2k )(ΓK,s (q/2))2k := q/2

Z

H

s−q ord p (x j1 )/2 · · · s−q ord p (x j2k )/2 µq,B,γ (dx)

q/2

= P ∂ j1 · · · P ∂ j2k µˆ q,B,γ (0) = ([ P Dq/2 ]2k µˆ (x))|x=0 .(e j1 , . . . , e j2k ), where ( P Dq/2 f (x)).e j := P ∂ j f (x). Therefore, q/2 (v) m2k (e j1 , . . . , e j2k )(ΓK,s (q/2))2k = (k!)−1 [ P Dq/2 ]2k [B(vsq (z), vsq (z)]k .(e j1 , . . . , e j2k ) = (k!)−1 ∑σ∈Σ2k Bσ( j1 ),σ( j2 ) · · · Bσ( j2k−1 ),σ( j2k ) , since γ = 0 and χγ (z) = 1, where Σk is the symmetric group of all bijective mappings σ of the set {1, . . . , k} onto itself, Bl, j := B(e j , eRl ), since Y ∗ = Y for Y = l2 (α, R). Therefore, for each B ∈ L1 and A ∈ L∞ we have H A(vq (x), vq (x))µq,B,0 (dx) = q/2 limN∈α ∑Nj=1 ∑Nk=1 A j,k m2 (e j , ek ) = Tr(AB), since Cs ⊂ Us and algebraically Cs is isomorphic with C. In particular for A = I and µq,tB,0 corresponding to the transition measure of ξ(t, ω) we get Formula (i) for a real time parameter, using µq,ln[χF (t)]B,0 we get Formula (ii) for a time parameter belonging to F, since ξ(t0 , ω) = 0 for each ω. 10. Corollary. Let H = K and ξ, B = 1, γ be as in Proposition 9, then   Z b Z s φ(t, ω)dt φ(t, ω)v2q (dξ(t, ω)) = M (i) M t∈[a,b]

a

for each a < b ∈ T with real time, where φ(t, ω) ∈ L(Ω, U, λ,C00 (T, R)) ξ ∈ L(Ω, U, λ, X0 (T, K)), (Ω, U, λ) is a probability measure space.

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Sergey V. Ludkovsky Proof. Since Z

t∈[a,b]

φ(t, ω)vs2q (dξ(t, ω)) =

N

∑ φ(t j , ω)vsq(ξ(t j+1 , ω) − ξ(t j , ω))

lim

max j (t j+1 −t j )→0 j=1

for λ-almost all ω ∈ Ω, since Cs ⊂ Us and Cs is algebraically isomorphic with C, then from the application of Formula 9.(i) to each vs2q (ξ(t j+1 , ω) − ξ(t j , ω)) and the existence of the limit by finite partitions a = t1 < t2 < · · · < tN+1 = b of the segment [a, b] it follows Formula 10.(i). 11. Definitions and Notes. Consider a pseudo-differential operator on H = c0 (α, K) such that (i) A = (−i)k bkj1 ,..., jk P ∂ j1 · · · P ∂ jk , ∑ 0≤k∈Z; j1 ,..., jk ∈α

where bkj1 ,..., jk ∈ R, P ∂ jk := P ∂1jk . If there exists n := max{k : bkj1 ,..., jk 6= 0, j1 , . . . , jk ∈ α}, then n is called an order of A, Ord(A), where P ∂ j is defined by Formula 9.(iii). If A = 0, then by definition Ord(A) = 0. If there is not any such finite n, then Ord(A) = ∞. We L suppose that the corresponding form A˜ on k Y k is continuous into C, where ˜ (ii) A(y) =−

∑

(−i)k bkj1 ,..., jk y j1 · · · y jk /lns,

0≤k∈Z; j1 ,..., jk ∈α

˜ y ∈ l2 (α, R) =: Y . If A(y) > 0 for each y 6= 0 in Y , then A is called strictly elliptic pseudodifferential operator. Let X be a complete locally K-convex space, let Z be a complete locally Us convex space. For 0 ≤ n ∈ R a space of all functions f : X → Z such that f (x) and ( P Dk f (x)).(y1 , . . . , yl(k) ) are continuous functions on X for each y1 , . . . , yl(k) ∈ {e1 , e2 , e3 , . . . }, l(k) := [k] + sign{k} for each k ∈ N such that k ≤ [n] and also for k = n is denoted by P C n (X, Z) and f ∈ P C n (X, Z) is called n times continuously pseudodifferentiable, where [n] ≤ n is an integer part of n, 1 > {n} := n − [n] ≥ 0 is a fractional T n part of n. Then P C ∞ (X, Z) := ∞ n=1 P C (X, Z) denotes a space of all infinitely pseudodifferentiable functions. Embed R into Cs and consider the function vs2 : Up → R ⊂ Cs , then for t = vs2 (θ), θ ∈ K ⊂ Up , put ∂t u(t, x) := limθ,K,θ∈K,vs2 (θ)→t P ∂θ u(vs2 (θ), x) for t ≥ 0, when it exists by S the filter of local subfields K ⊂ Cp , which is correct, since vs2 (Up ) = [0, ∞), K⊂Cp K is dense in Cp , ΓCp = (0, ∞) ∩ Q. 12. Theorem. Let A be a strictly elliptic pseudo-differential operator on H = c0 (α, K), card(α) ≤ ℵ0 , and let t ∈ T = [0, b] ⊂ R. Suppose also that u0 (x − y) ∈ L(H, Bco(H), µt A˜ , Us ) for each marked y ∈ H as a function by x ∈ H, u0 (x) ∈ Ord(A) (H, U ). Then the non-Archimedean analog of the Cauchy problem PC s (i) ∂t u(t, x) = Au,

u(0, x) = u0 (x)

has a solution given by (ii) u(t, x) =

Z

H

u0 (x − y)µt A˜ (dy), ˜

s

where µt A˜ is a Ks -valued measure on H with a characteristic functional µˆ t A˜ (z) := st A(v2 (z)) .

Non-Archimedean Valued Measures

137 ˜

s

Proof. In accordance with § § 2 and 11 we have Y = l2 (α, R). The function st A(v2 (z)) is continuous on H ֒→ H ∗ for each t ∈ R such that the family H of continuous K-linear functionals on H separates points in H. In view of Theorem 2.30 above it defines a tight measure on H for each t > 0. The functional A˜ on each ball of radius 0 < R < ∞ in Y is ˜ L [span (e ,...,e )]k , when n tends to the infinity, since A˜ a uniform limit of its restrictions A| k

K

n

1

is continuous on k Y k . Since u0 (x − y) ∈ L(H, Bco(H), µt A˜ , Us ) and a space of cylindrical functions is dense in the latter Banach space over Us , then in view of theorems about the isomorphism of K with Kˆ. for a locally compact field and the Schikhof’s theorem about the isomorphism L(G) ≃ C∞ (Gˆ.K ) formulated above (see also Theorems 9.14 and 9.21 in [Roo78]) and the Fubini theorem it follows that limP→I FPx u0 (Px))ˆµt A˜ (y + Px) converges in L(H, Bco(H), µt A˜ , Us ) for each t, since µt1 A˜ ∗ µt2 A˜ = µ(t1 +t2 )A˜ for each t1 , t2 and t1 + t2 ∈ T , where P is a projection on a finite dimensional over K subspace HP := P(H) in H, HP ֒→ H, P tends to the unit operator I in the strong operator topology, FPx u0 (Px) denotes a Fourier transform by the variable Px ∈ HP . ˜ s (x))v(t, x)lns, We consider now the function v := Fx (u), then ∂t v(t, x) = −A(v 2 ˜ s (x)) t A(v −1 consequently, v(t, x) = v0 (x)s 2 . From u(t, x) = Fx (v(t, x)), where Fx (u(t, x)) R −1 = limn→∞ Fx1 ,...,xn u(t, x). Therefore, u(t, x) = u0 (x)∗[Fx (ˆµt A˜ )] = H u0 (x−y)µt A˜ (dy), since u0 (x − y) ∈ L(H, Bco(H), µt A˜ , Us ) and µt A˜ is the tight measure on Bco(H). 14. Note. In the particular case of Ord(A) = 2 and A˜ corresponding to the Laplace ˜ operator, that is, A(y) = ∑l, j gl, j yl y j , Equation 12.(i) is (the non-Archimedean analog of) the heat equation on H. For Ord(A) < ∞ the form A˜ 0 (y) corresponding to sum of terms with k = Ord(A) in Formula 11.(ii) is called the principal symbol of operator A. If A˜ 0 (y) > 0 for each y 6= 0, then A is called an elliptic pseudo-differential operator. Evidently, Theorem 13 is true for elliptic A of Ord(A) < ∞. 15. Remark and Definitions. Let linear spaces X over K and Y over R be as in § 4 and B be a symmetric nonnegative definite (bilinear) operator on a dense R-linear subspace DB,Y in Y ∗ . A quasi-measure µ with a characteristic functional L

µˆ (ζ, x) := sζB(vq (z),vq (z)) χγ (z) s

s

for a parameter ζ ∈ Cs with Re(ζ) ≥ 0 defined on Dq,B,X we call an Us -valued (nonArchimedean analog of Feynman) quasi-measure and we denote it by µq,ζB,γ also, where Dq,B,X := {z ∈ X ∗ : there exists j ∈ ϒ such that z(x) = z j (φ j (x)) ∀x ∈ X, vsq (z) ∈ DB,Y }. 16. RProposition. Let X = Dq,B,X and B be positive definite, then for each function f (z) := X χz (x)ν(dx) with an Us -valued tight measure ν of finite norm and each Re(ζ) > 0 there exists Z Z (P) f (Pz)µζB (dz) (i) f (z)µζB (dz) = lim P→I X

X

=

Z

X

s(ζB(vq (z),vq (z))) χγ (z)ν(dz),

where µ(P) (P−1 (A)) := µ(P−1 (A)) for each A ∈ Bco(XP ), P : X → XP is a projection on a K-linear subspace XP , a convergence P → I is considered relative to a strong operator topology.

138

Sergey V. Ludkovsky

Proof. From the use of the projective limit decomposition of X and Theorem 2.37 it follows, that there exists Z

(ii)

f (z)µζB (dz) = lim

Z

(P)

f (Pz)µζB (dz).

P→I X

X

Then for each finite dimensional over K subspace XP (iii)

Z

(P)

X

f (Pz)µζB (dz) =

Z

{sζB(vq (z),vq (z))) χγ (z)}|XP νXP (dz), s

XP

s

since ν is tight and hence each νXP is tight. Each measure ν j is tight, then due to Lemma 2.3 and § 2.5 above there exists the limit lim

Z

s

P→I XP

=

{sζB(vq (z),vq (z)) χγ (z)}|XP νXP (dz)

Z

s

sζB(vq (z),vq (z))) χγ (z)ν(dz). s

X

s

17. Proposition. If conditions of Proposition 16 are satisfied and (i)

f (Px) ∈ L(XP , Bco(wXP ), Us )

for each finite dimensional over K subspace XP in X and (ii)

lim sup | f (x)| = 0,

R→∞ |x|≤R

then Formula 16(i) is accomplished for ζ with Re(ζ) = 0, where wXP is a nondegenerate Ks -valued Haar measure on XP . Proof. In view of Theorem 2.37 for the consistent family of measures XP { f (Px)µq,iB,γ (dPx) : P} (see § 2.36) there exists a measure on (X, R ), where projection operators P are associated with a chosen basis in X. The finite dimensional over K distriP bution µXq,iB,γ /wXP (dx) = F −1 (ˆµq,iB,γ )|XP ) is in C∞ (XP , Us ) due to Theorem 9.21 [Roo78], since µˆ ∈ L(XP , Bco(Xp ), wXP , Us ). In view of Condition 17.(i, ii) above and the Fubini theorem and using the Fourier-Stieltjes transform we get Formulas 16(ii, iii). From the taking the limit by P → I Formula 16.(i) follows. This means that µq,ζB,γ exists in the sense of distributions. 18. Remark. Put (i)

F

Z

X

f (x)µq,iB,γ (dx) := lim

Z

ζ→i X

f (x)µq,ζB,γ (dx)

if such limit exists. If conditions of Proposition 17 are satisfied, then ψ(ζ) := X f (x)µq,ζB,γ (dx) is the pseudo-differentiable of order 1 function by ζ on the set {ζ ∈ Cs : Re(ζ) > 0} and it is continuous on the subset {ζ ∈ Cs : Re(ζ) ≥ 0}, consequently, R

(ii)

F

Z

X

f (x)µq,iB,γ (dx) =

Z

s

X

s

s{iB(vq (x),vq (x))} χγ (x)ν(dx).

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139

Above non-Archimedean analogs of Gaussian measures with specific properties were defined. Nevertheless, there do not exist usual Gaussian Ks -valued measures on nonArchimedean Banach spaces. 19. Theorem. Let X be a Banach space of separable type over a locally compact non-Archimedean field K. Then on Bco(X) there does not exist a nontrivial Ks -valued (probability) usual Gaussian measure. Proof. Let µ be a nontrivial usual Gaussian Ks -valued measure on Bco(X). Then by the definition its characteristic functional µˆ must be satisfying Conditions 2.5.(3, 5) Us -valued function and lim|y|→∞ µˆ (y) = 0 for each y ∈ X ∗ \ {0}, where X ∗ is the topological conjugate space to X of all continuous K-linear functionals f : X → K. Moreover, there exist a K-bilinear functional g and a compact non-degenerate K-linear operator T : X ∗ → X ∗ with ker(T ) = {0} and a marked vector x0 ∈ X such that µˆ x0 (y) = f (g(Ty, Ty)) for each y ∈ X ∗ , where µx0 (dx) := µ(−x0 + dx), x ∈ X. Since K is locally compact, then X ∗ is nontrivial and separates points of X (see [NB85, Roo78]). Each one-dimensional over K projection of a Gaussian measure is a Gaussian measure and products of Gaussian measures are Gaussian measures, hence convolutions of Gaussian measures are also Gaussian measures. Therefore, µˆ x0 : X ∗ → Us is a nontrivial character: µˆ x0 (y1 + y2 ) = µˆ x0 (y1 )ˆµx0 (y2 ) for each y1 and y2 in X ∗ . If char(K) = 0 and K is a non-Archimedean field, then there n exists a prime number p such that Qp is the subfield of K. Then µˆ (pn y) = (ˆµ(y)) p for each n ∈ Z and y ∈ X ∗ \ {0}, particularly, for n ∈ N tending to the infinity we have n limn→∞ pn y = 0 and limn→∞ µˆ x0 (pn y) = 1, limn→∞ µˆ x0 (y)) p = 0, since s 6= p are primes, limn→∞ µˆ x0 (p−n y) = 0 and |ˆµx0 (y)| < 1 for y 6= 0. This gives the contradiction, hence K can not be a non-Archimedean field of zero characteristic. Suppose that K is a non-Archimedean field of characteristic char(K) = p > 0, then K is isomorphic with the field of formal power series in variable t over a finite field Fp . Therefore, µˆ x0 (py) = 1, but µˆ x0 (y) p 6= 1 for y 6= 0, since limn→∞ µˆ x0 (t −n y) = 0. This contradicts the fact that µˆ x0 need to be the nontrivial character, consequently, K can not be a non-Archimedean field of nonzero characteristic as well. It remains the classical case of X over R or C, but the latter case reduces to X over R with the help of the isomorphism of C as the R-linear space with R2 . 20. Theorem. Let µq,B,γ and µq,B,δ be two q-Gaussian Ks -valued measures. Then µq,B,γ is equivalent to µq,B,δ or µq,B,γ ⊥ µq,B,δ according to vsq (γ − δ) ∈ B1/2 (DB,Y ) or not. The measure µq,B,γ is orthogonal to µg,B,δ , when q 6= g. Two measures µq,B,γ and µg,A,δ with positive definite nondegenerate A and B are either equivalent or orthogonal. 21. Theorem. The measures µq,B,γ and µq,A,γ are equivalent if and only if there exists a positive definite bounded invertible operator T such that A = B1/2 T B1/2 and T − I ∈ L2 (Y ∗ ). Proof. Using the projective limit reduce consideration to the Banach space X. Let z ∈ X be a marked vector and Pz be a projection operator on zK such that Pz2 = Pz , z = ∑ j z j e j , then the characteristic functional of the projection µzK q,B,γ of µq,B,γ has the form

[(∑i, j Bi, j vq (zi )vq (z j ))v2q (ξ)] µˆ zK χγ(z) (ξ) for each vector x = ξz, where each z j and ξ ∈ K, q,B,γ = s 2 s s since v2q (ξ) = (vq (ξ)) . Choose a sequence { n z : n} in X such that it is the orthonormal basis in X and the operator G : X → X such that G n z = n a n z with n a 6= 0 for each n ∈ N and there exists G−1 : G(X) → X such that it induces the operator C on a dense subspace s

s

s

140

Sergey V. Ludkovsky

D (Y ) in Y such that CBC : Y → Y is invertible and kCBCk and k(CBC)−1 k ∈ [|π|, |π|−1 ]. n n Then µq,A,γ (dx)/µq,B,γ (dx) = limn→∞ [µVq,A,γ (dxn )/λVn (dxn )][µVq,B,γ (dxn )/λVn (dxn )]−1 , where Vn := spanK ( j z : j = 1, . . . , n), xn ∈ Vn . Consider xn = G−1 (yn ), where yn ∈ G(Vn ), then n n [µVq,B,γ (G−1 dyn )/λVn (G−1 dyn )] and [µVq,B,γ (G−1 dyn )/λVn (G−1 dyn )]−1 are in Vn V −1 n −1 n n L(λ (G dy )) for each n such that there exists m ∈ N for which k[µq,B,γ (G dy )/ n λVn (G−1 dyn )]k and k[µVq,B,γ (G−1 dyn )/λVn (G−1 dyn )]−1 k ∈ [|π|, |π|−1 ] for each n > m, where k ∗ k is taken in L(λVn (G−1 dyn )). Then NµVn (yn ) ∈ [|π|, |π|−1 ] for each n > m. Then q,CBC,γG−1

the existence of µq,A,γ (dx)/µq,B,γ (dx) ∈ L(µq,B,γ ) is provided by using operator G and the consideration of characteristic functionals of measures, Theorem 3.5 and the fact that the Fourier-Stieltjes transform F is the isomorphism of Banach algebras L(K, Bco(K), v, Us ) with C∞ (K, Us ), where v denotes the Haar normalized by v(B(K, 0, 1)) = 1 Ks -valued measure on K. If g 6= q then the measure µq,B,γ is orthogonal to µg,B,δ , since lim

sup |(µq,B,γ )Xn /(µg,B,δ )Xn |(x) = 0

R>0,R+n→∞ x∈X c

R,n

c := for each q > g due to Formula 4.(ii), where Xn := spanK (em : m = n, n + 1, . . . , 2n), XR,n Xn \ B(Xn , 0, R) , (µq,B,γ )Xn is the projection of the measure µq,B,γ on Xn . Each term β j in Theorem 3.5 is in [0, 1] ⊂ R, consequently, the product in this theorem is either converging to a positive constant or diverges to zero, hence two measures µq,B,γ and µg,A,δ are either equivalent or orthogonal.

2.7.

Comments

1. In the article of W. Schikhof [Sch71] it was investigated the non-Archimedean analog of the Radon-Nikodym theorem. Let (X, R , µ) be a measure space with a K-valued measure µ and a covering ring R of X, where K is a non-Archimedean field complete relative to its nontrivial uniformity. If 1 ∈ L(X, R , µ, K), then Nµ is bounded on X. Let Ω := {U ⊂ X : fChU ∈ L(X, R , µ, K) for each f ∈ L(X, R , µ, K)}. 2. Definition. Let ψ : Ω → K be a function and µ be an integral corresponding to a measure µ and denoted by the same letter. Suppose x ∈ X, a ∈ K, r ∈ R. (1). If for each ε > 0 there exists a neighborhood V ∈ R of x such that for all U ⊂ V , U ∈ Ω the inequality |a − ψ(U)| < ε is satisfied, then we write LIMU→x ψ(U) = a. (2). If for each ε > 0 there exists a neighborhood V of x such that for all U ⊂ V , U ∈ Ω, with |µ(U)| ≥ cNµ (x) we have |a − ψ(U)| < ε, then we write LIMµ,c;U→x ψ(U) = a. (3). LIMµ,U→x ψ(U) = a means LIMµ,c;U→x ψ(U) = a for each c ∈ (0, 1). (4). LIMU→x |ψ(U)| = r means, that for each ε > 0 there exists a neighborhood V ∈ R of x such that r − ε ≤ sup{|ψ(U)| : U ∈ Ω, U ⊂ V } ≤ r + ε. 3. Theorem. Let µ be an integral on L(X, R , µ, K) and let f ∈ L(X, R , µ, K) and x ∈ X. Then (i). LIMU→x (µ( fChU ) − f (x)µ(U)) = 0. If Nµ (x) > 0, then LIMµ,U→x µ( fChU )µ(U)−1 = f (x). 4. Theorem. Let µ and ν be K-valued measures on (X, R ). Then the following conditions are equivalent:

Non-Archimedean Valued Measures

141

(i). There exists g ∈ L(µ) such that f g ∈ L(µ) for each f ∈ L(ν) and ν( f ) = µ( f g). (ii). For each x ∈ X for which Nµ (x) > 0 there exists q(x) = LIMµ,U→x ν(U)µ(U)−1 . Moreover, every µ-null set is ν-null. (iii). There exists a function h : X → K such that for each x ∈ X: LIMU→a [ν(U) − h(x)µ(U)] = 0. Further, g, q, h are uniquely determined up to µ-null functions and g = q = h µ-almost everywhere. Finally, Nν = |g|Nµ . 5. Theorem. Let φ : X → Y be a surjective homeomorphism of (X, R X ) on (Y, R Y ) such that the mapping f 7→ f ◦ φ is a bijection of L(Y, R Y , ν, K) on L(X, R X , µ, K). Suppose that Conditions (1, 2) below are satisfied. (1). For x ∈ X with Nµ (x) > 0 there exists g(x) = LIMµ,U→x µ(U)−1 ν(φ(U)). (2). For each x ∈ X, Nµ (x) = 0 implies Nν (φ(x)) = 0. Then g is µ-integrable and for every f ∈ L(ν) we have ν( f ) = µ(( f ◦ φ)g). 6. Remark. In § § 2.31, 3.9 and 3.12 the specific definitions of absolute continuity and quasi-invariance of Ks -valued measures were given in view of Radon-Nikodym Theorem 4. The latter Theorem 5 serves for substitution of variables in the integral. The fields Us used above were investigated, for example, in [Dia84] (see also [Esc95, Roo78] and references therein). 7. Definitions. Let (K j : j ∈ J) be a family of fields supplied with a non-Archimedean normalization. Consider a sub-ring ∏ j∈J K j of the product ring ∏ j∈J K j formed by elements a = (a j : j ∈ J) ∈ ∏ j∈J K j such that sup j∈J |a j | < ∞. Let U be an ultrafilter on the set J. We define an ultra-metric semi-normalization on ∏ j∈J K j such that |a| := limU |a j |. The subset formed of all a ∈ ∏ j∈J K j with |a| = 0 forms the ideal denoted by J . We say that the quotient ring [∏ j∈J K j ]/J supplied with the quotient absolute value is the ultra-product of fields K j and it is denoted by ∏ j∈J K j/U also. Denote also by a and |a| elements in ∏ j∈J K j/U and their absolute values. Remind that an ultrafilter U on a set J is ω-incomplete, if there exists a sequence (Xn : T / n ≥ 0), Xn ∈ U , Xn+1 ⊂ Xn for each n ≥ 0 such that n≥0 Xn = 0. 8. Theorem. Let (K j : j ∈ J) be a family of fields with discrete valuations v j such that v j (K j ) = Z and let (ρ j : j ∈ J) be a family of real numbers such that 0 < ρ j < 1 for each j. v (a )

Consider in each K j an absolute value |a j | := ρ j j j . (1). If limU ρ j = 0, then the absolute value in ∏ j∈J K j/U is trivial. (2). If limU ρ j = 1, then the field K := ∏ j∈J K j/U has the dense normalization such that ΓK = (0, ∞). Moreover, K is ω-incomplete and spherically complete. (3). If 0 < limU ρ j = ρ < 1, then ∏ j∈J K j/U is the field with the discrete valuation. (4). If each K j is algebraically complete, then K is algebraically complete. 9. Proposition. Let L be a field with a discrete normalization group ΓL . Consider a family (K j : j ∈ J) of all finite extensions of L such that [K j : L] = [k j : l] = n j , where k j is the residue class field of K j and l is that of L. (1). The field K := ∏ j∈J K j/U is a complete field with a discrete valuation and with the residue class field k := ∏ j∈J k j/U . (2). If limU n j = ∞ and if l is perfect, then the field K := ∏ j∈J K j/U is the transcendental extension of Lo := ∏ j∈J L/U .

Chapter 3

Algebras of Real Measures on Groups 3.1.

Introduction

Besides Banach spaces quasi-invariant measures were constructed on non-locally compact topological groups. For example, on a group of diffeomorphisms they were constructed for real locally compact manifolds M in [Kos94, Sha89] and for non-locally compact real or non-Archimedean manifolds M in [Lud96, Lud99t, Lud99r, Lud0348]. Such groups are also Banach manifolds or strict inductive limits of their sequences. Then on a real and non-Archimedean wrap (particularly loop) groups and semigroups of families of mappings from one manifold into another they were elaborated in [Lud98s, Lud00d, Lud02b, Lud00a, Lud08]. Then each Banach space over a locally compact field supplies an example of the additive group and quasi-invariant real-valued measures on it were described in Chapter 1. On real and non-Archimedean Banach-Lie groups quasi-invariant measures were constructed in [DS69, Lud0348]. This chapter is devoted to the investigation of properties of quasi-invariant measures that are important for analysis on topological groups and for construction irreducible representations [Kos94, Ner88]. Algebras of quasi-invariant measures and functions are defined and studied on topological groups. The following properties are investigated: (1) convolutions of measures and functions, (2) continuity of functions of measures, (3) non-associative algebras generated with the help of quasi-invariant measures. The theorems given below show that many differences appear to be between locally compact and non-locally compact groups. The topological Hausdorff groups considered below are supposed to have structure of Banach manifolds over the corresponding fields if something another is not specified.

3.2.

Algebras of Measures and Functions

1. Definitions. (a). Let G be a Hausdorff separable topological group. A real (or complex) Radon measure µ on A f (G, µ) is called left-quasi-invariant (or right) relative to a dense

144

Sergey V. Ludkovsky

subgroup H of G, if µφ (∗) (or µφ (∗)) is equivalent to µ(∗) for each φ ∈ H, where B f (G) is the Borel σ-field of G, A f (G, µ) is its completion by µ, µφ (A) := µ(φ−1 A), µφ (A) := µ(Aφ−1 ) for each A ∈ A f (G, µ), ρµ (φ, g) := µφ (dg)/µ(dg) (or ρ˜ µ (φ, g) := µφ (dg)/µ(dg)) denote a left (or right) quasi-invariance factor. We assume that a uniformity τG on G is such that τG |H ⊂ τH , (G, τG ) and (H, τH ) are complete. We suppose also that there exists an open base in e ∈ H such that their closures in G are compact (such pairs exist for loop groups and groups of diffeomorphisms and Banach-Lie groups). We denote by Ml (G, H) (or Mr (G, H)) a set of left-( or right) quasi-invariant measures on G relative to H with a finite norm kµk := supA∈A f (G,µ) |µ(A)| < ∞. (b). Let LHp (G, µ, C) for 1 ≤ p ≤ ∞ denotes the Banach space of functions f : G → C such that fh (g) ∈ L p (G, µ, C) for each h ∈ H and k f kLHp (G,µ,C) := sup k fh kL p (G,µ,C) < ∞, h∈H

where fh (g) := f (h−1 g) for each g ∈ G. For µ ∈ Ml (G, H) and ν ∈ M(H) let (ν ∗ µ)(A) :=

Z

H

µh (A)ν(dh) and (q˜∗ f )(g) :=

Z

f (hg)q(h)ν(dh) H

be convolutions of measures and functions, where M(H) is the space of Radon measures on H with a finite norm, ν ∈ M(H) and q ∈ Ls (H, ν, C), that is Z

H

1/s =: kqkLs (H,ν,C) < ∞ for 1 ≤ s < ∞. |q(h)|s |ν|(dh)

2. Lemma. The convolutions are continuous C-linear mappings ∗ : M(H) × Ml (G, H) → Ml (G, H) and 1 1 ∗˜ : L1 (H, ν, C) × LH (G, µ, C) → LH (G, µ, C).

Proof. It follows immediately from the definitions, Fubini theorem and because dµ (h, g) ∈ L1 (H × G, ν × µ, C). In fact one has, kν ∗ µk ≤ kνk × kµk, kq˜∗ f kLH1 (G,µ,C) ≤ kqkL1 (H,ν,C) × k f kLH1 (G,µ,C) . 3. Definition. For µ ∈ M(G) its involution is given by the following formula: µ∗ (A) := µ(A−1 ), where b¯ denotes complex conjugated b ∈ C, A ∈ A f (G, µ). 4. Lemma. Let µ ∈ Ml (G, H) and G and H be non-locally compact with structures of Banach manifolds. Then µ∗ is not equivalent to µ. Proof. Let T : G → T G be the tangent mapping. Then µ induces quasi-invariant measure λ from an open neighborhood W of the unit e ∈ G on a neighborhood of the zero section V in Te G and then it has an extension onto the entire Te G. Let at first Te G be a Hilbert space. Put Inv(g) = g−1 then T ◦ Inv ◦ T −1 =: K on V is such that there is not any operator B of trace class on Te G such that M˜ λ ⊂ B1/2 Te G and KTe G ⊂ M˜ λ , where Re(1 − θ(z)) → 0 for (Bz, z) → 0 and z ∈ Te G, θ(z) is the characteristic functional of λ, M˜ λ is a set of all x ∈ Te G such that λx is equivalent to λ (see Theorem 19.1 [Sko74]). Then using theorems

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for induced measures from a Hilbert space on a Banach space [DF91, Kuo75], we get the statement of Lemma 4. 5. Lemma. For µ ∈ Ml (G, H) and 1 ≤ p < ∞ the translation map (q, f ) → fq (g) is continuous from H × LHp (G, µ, C) into LHp (G, µ, C). Proof. For metrizable G in view of the Lusin theorem (2.3.5 in [Fed69]) and definitions of τG and τH for each ε > 0 there are a neighborhood V ∋ e in H and compacts K1 and K in G such that the closure clGV K1 =: K2 is compact in G with K2 ⊂ K, a restriction f |K2 is continuous, (|˜µ| + |µ|)(G \ K2 ) < ε, where µ˜ (dg) := f (g)µ(dg). 6. Proposition. For a probability measure µ ∈ M(G) there exists an approximateR unit, that is a sequence of non-negative continuous functions ψi : G → R such that G ψi (g)µ(dg) = 1 and for each neighborhood U ∋ e in G there exists i0 such that supp(ψi ) ⊂ U for each i > i0 . Proof. A finite union of compact subsets in G is compact. So in view of the Radon property of µ for each b > 0 and for each symmetric U = U −1 neighborhood U of e in G there exist a compact set Cb in G such that e ∈ Cb and µ(U \Cb ) < bµ(U) and µ(G \Cb ) < b. The group G has the structure of the Banach manifold also, hence its base of open neighborhoods of the unit element e is countable. Choose such base {U j : j} with U j = U j−1 and cl(U j+1 ) ⊂ U j for each j ∈ N, with T − j and U = U and the corresponding comj j U j = {e}. Then we get the sequence b j = 2 pact subsets Cb j in G. Take continuous non-negative functions φ j with supp(φ j ) ⊂ U j so that φ j (g) = k j for each g ∈ Cb j+1 ∩U j+1 , where k j ≥ 1 is a positive constant. This is possible due to the Uryson theorem (see § 1.5.10 in [Eng86]). This theorem states that if A and B are two disjoint closed subsets in a normal topological space S, then there exists a continuous function f : S → [0, 1] such that f (A) = {1} and f (B) = {0}. Therefore, 0 < µ(φ j ) − k j µ(Cb j+1 ∩U j+1 ) < k j 2− j−1 , so there exists a positive constant multiplier h j so that ψ j := h j φ j satisfies the equality µ(ψ j ) = 1, certainly supp(ψ j ) ⊂ Uk for each j ≥ k by the given construction. 7. Proposition. If (ψi : i ∈ N) is an approximate unit in H relative to a probability 1 (G, µ, C) norm, where µ ∈ M (G, H), measure ν ∈ M(H), then limi→∞ ψi ∗˜ f = f in the LH l 1 f ∈ LH (G, µ, C). Proof. Given b > 0 we choose a neighborhood U of e in G such that k f (x) − f (e)kLH1 < 1 into L1 . If j is large b for each x ∈ U. The mapping x 7→ f (xy) is continuous from H × LH H −1 enough so that supp(ψ j ) ⊂ V , where V = V is open and symmetric neighborhood of e in G and V 2 ⊂ U, then

Z

Z





f (yx)ψ j (y)ν(dy) − f (e) = [ f (yx)ψ j (y) − f (e)]ν(dy)

H

≤

1 LH

Z

U∩H

H

ψ j (y)k f (yx) − f (e)kLH1 ν(dy) ≤ b

1 LH

Z

U∩H

ψ j (y)ν(dy) ≤ b,

1 into L1 . In view of Lemmas 2, 5 we where the convolution with ψ j is the operator from LH H get the statement of this proposition. q 8. Lemma. Suppose g ∈ LH (G, µ, C) and (gx |H ) ∈ Lq (H, ν, C) for each x ∈ G, f ∈ L p (H, ν, C) with 1 < p < ∞, 1/p + 1/q = 1, where gx (y) := g(yx) for each x and y ∈ G. Let 1 (G, µ, C) and µ and ν be probability measures, µ ∈ Ml (G, H), ν ∈ M(H). Then f ∗˜ g ∈ LH

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there exists a function h : G → C such that h|H is continuous, h = f ∗˜ g µ-a.e. on G and h vanishes at ∞ on G. Proof. In view of Fubini theorem and H¨older inequality we have k f ∗˜ gkLH1 (G,µ,C) = sup

Z Z

| f (y)| × |g(z)|ν(dy)µ((ys)−1 dz) ≤

s∈H G H

sup s∈H

Z Z

G H

1/p 1/q  Z Z p −1 ≤ | f (y)| ν(dy)µ((ys) dz) × |g(z)| ν(dy)µ((ys) dz) −1

q

G H

k f kL p (H,ν,C) × kgkLHq (G,µ,C) × ν(H)µ(G).

The equation α f (φ) := H f (y)φ(y)ν(dy) defines a continuous linear functional on −1 −1 ) =: w(s, x) of two ˜ Lq (H, ν, C). In view of Lemma 5 the function α f (g(sx) ) =: h((sx) variables s and x is continuous on H × H for s, x ∈ H, since the mapping (s, x) 7→ (sx)−1 is continuous from H × H into H. By Fubini theorem (see § 2.6.2 in [Fed69]) R

Z

h(y)ψ(y)µ(dy) =

Z Z

f (y)g(yx)ψ(x)ν(dy)µ(dx)

G H

G

=

Z

H

for each ψ ∈ L p (G, µ, C), since Z Z

G H

 Z g(yx)ψ(x)µ(dx) ν(dy) f (y) G

| f (y)g(yx)ψ(x)| |ν|(dy) |µ|(dx) < ∞,

˜ −1 ). Here ψ is where |ν| denotes the variation of the real-valued measure ν, h(y) := h(y arbitrary in L p (G, µ, C), from this it follows, that µ({y : h(y) 6= ( f ∗˜ g)(y), y ∈ G}) = 0, since h and ( f ∗˜ g) are µ-measurable functions due to Fubini theorem and the continuity of the composition and the inversion in a topological group. In view of Lusin theorem (see § 2.3.5 in [Fed69]) for each ε > 0 there are compact subsets C ⊂ H and D ⊂ G and functions q f ′ ∈ L p (H, ν, C) and g′ ∈ LH (G, µ, C) with closed supports supp( f ′ ) ⊂ C, supp(g′ ) ⊂ D such that clGCD is compact in G, k f ′ − f kL p (H,ν,C) < ε and kg′ − gkLHq (G,µ,C) < ε, since by the supposition of § 1 the group H has the base BH of its topology τH , such that the closures clGV are compact in G for each V ∈ BH . From the inequality |h′ (x) − h(x)| ≤ (k f kL p (H,ν,C) + ε)ε + εkgkLHq (G,µ,C) it follows that for each δ > 0 there exists a compact subset K ⊂ G with |h(x)| < δ for each x ∈ G \ K, where h′ (x−1 ) := α f ′ (g′ x ). 9. Proposition. Let A, B ∈ A f (G, µ), µ and ν be probability measures, µ ∈ Ml (G, H), ν ∈ M(H). Then the function ζ(x) := µ(A ∩ xB) is continuous on H and ν(yB−1 ∩ H) ∈ L1 (H, ν, C). Moreover, if µ(A)µ(B) > 0, µ({y ∈ G : yB−1 ∩ H ∈ A f (H, ν) and ν(yB−1 ∩ H) > 0}) > 0, then ζ(x) 6= 0 on H.

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Proof. Let gx (y) := ChA (y)ChB (x−1 y), then gx (y) ∈ LH (G, µ, C) for 1 < q < ∞, where ChA (y) is the characteristic function of A. In view of Propositions 6 and 7 there exists 1 (G, µ, C). In view of § 7 and Lemma 8 ζ(x)| is continuous. There limi→∞ ψi ∗˜ gx = gx in LH H is the following inequality: q

1≥

Z

µ(A ∩ xB)ν(dx) = H

Z Z

H G

chA (y)chB (x−1 y)µ(dy)ν(dx).

In view of Fubini theorem there exists Z

H

ChB (x−1 y)ν(dy) = ν((yB−1 ) ∩ H) ∈ L1 (G, µ, C), hence Z

µ(A ∩ xB)ν(dx) = H

Z

G

ν(yB−1 ∩ H)chA (y)µ(dy).

10. Corollary. Let A, B ∈ A f (G, µ), ν ∈ M(H) and µ ∈ Ml (G, H) be probability measures. Then denoting IntH V the interior of a subset V of H with respect to τH , one has / when (i) IntH (AB) ∩ H 6= 0, µ({y ∈ G : ν(yB ∩ H) > 0}) > 0; (ii) IntH (AA−1 ) ∋ e, when µ({y ∈ G : ν(yA−1 ∩ H) > 0}) > 0. Proof. AB ∩ H ⊃ {x ∈ H : µ(A ∩ xB−1 ) > 0}. 10.1. Corollary. Let µ ∈ Ml (G, H), ρµ (h, z) ∈ L1 (H, ν, C) × L1 (G, µ, C), then ρµ (h, z) is continuous ν × µ-a.e. on H × G. Proof. Recall the Lusin’s theorem. If φ is a Borel regular non-negative measure on a metric space X or a Radon measure on a locally compact Hausdorff space X, a function f is φ-measurable with values in a separable metric space Y , A ⊂ X and A is φ-measurable with φ(A) < ∞, b > 0, then A contains a closed or compact respectively subset Cb such that φ(A \ Cb ) < b and the restriction f |Cb of f on Cb is continuous (see also Theorem 2.3.5 in [Fed69]). In view of the co-cycle condition ρµ (φψ, g) := µφψ (dg)/µ(dg) = (µφψ (dg)/µφ (dg))(µφ (dg)/µ(dg)) = ρµ (ψ, φ−1 g)ρµ (φ, g) on ρµ and Corollary 10 above and the Lusin’s theorem for each ε > 0 the quasi-invariance factor ρµ (h, g) is continuous on H × Aε , where Aε is a compact subset in G, but Hε := H ∩ Gε is a neighborhood of e in H, where Gε := Aε ◦ A−1 ε . Since H is dense in G, then S S∞ µ(G \ ∞ h G ) = 0, where {h : j ∈ N} is a countable subset in G. Therefore, j j 1/n n=1 j=1 ρµ (h, z) is continuous ν × µ-a.e. on H × G. 10.2. Corollary. Let G be a locally compact group, ρµ (h, z) ∈ L1 (G × G, µ × µ, C), then ρµ (h, z) is µ × µ-a.e. continuous on G × G.

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11. Corollary. Let G = H. If µ ∈ Ml (G, H) is a probability measure, then G is a locally compact topological group. Proof. Let us take ν = µ and A = C ∪ C−1 , where C is a compact subset of G with µ(C) > 0, whence µ(yA) > 0 for each y ∈ G and inevitably IntG (AA−1 ) ∋ e. 12. Lemma. Let µ ∈ Ml (G, H) be a probability measure and G be non-locally compact. Then µ(H) = 0. Proof. In the particular case of a real Hilbert space X if a non-negative quasi-invariant measure µ on it is such Rthat µ(L) = 0 for each finite dimensional subspace, then µ(Mµ ) = 0, where Mµ := {a ∈ X : X ρ˜ µ (a, x)µ(dx) = µ(X)} (see Theorem 19.2 [Sko74]). On the other hand for each real Banach space Y their exists a dense Hilbert subspace X a quasi-invariant measure µ on which indices a quasi-invariant measure ν on the initial Banach space Y (see [GV61] and § I.4 in [Kuo75]). On the other hand, a quasi-invariant measure on G relative to left shifts Lh g := hg with h ∈ G′ from the dense subgroup G′ , where g ∈ G, induces a quasi-invariant measure on its tangent space Te G, since G has the structure of the smooth Banach manifold (see also [Kl82, Bou76]). Therefore the statement of this lemma follows from and Theorem 3.25 of Chapter 1 and the proof of Lemma 4, since the embedding Te H ֒→ Te G is a compact operator in the nonArchimedean case and of trace class in the real case. See also the papers about construction of quasi-invariant measures on the considered here groups [DS69, Sha89, Kos94, Lud96, Lud99t, Lud98s, Lud99r, Lud00a, Lud00d, Lud02b]. Indeed, the measure µ on G is induced by the corresponding measure ν on a Banach space Z for which there exists a local diffeomorphism A : W → V , where W is a neighborhood of e in G and V is a neighborhood of 0 in Z. The measure µ on G is quasi-invariant relative to H. Therefore, the measure ν on U is quasi-invariant relative to the action of elements ψ ∈ W ′ ⊂ W ∩ H due to the local diffeomorphism A, that is, νφ is equivalent to ν for each φ := AψA−1 , where AW ′ A−1U ⊂ V , W ′ is an open neighborhood of e in H and U is an open neighborhood of 0 in Z, νφ (E) := ν(φ−1 E), φ is an operator on Z such that it may be non-linear. The quasi-invariance factor ρν (φ, v) has expressions through |det(φ′ )| and the quasi-invariance factor qν (z, x) relative to linear shifts z ∈ Z ′ given by theorems from § 26 [Sko74] in the real case and Theorem 3.25 of Chapter 1 in the non-Archimedean case: νφ (dx)/ν(dx) = |det{φ′ (φ−1 (x))}|qν (x − φ−1 (x), x), where x ∈ U, φ = AψA−1 , ψ ∈ W ′ . Then (AψA−1 v − v) ∈ Z ′ for each v ∈ V and ψ ∈ W ′ , where ν on Z is quasi-invariant relative to shifts on vectors z ∈ Z ′ and there exists a compact operator in the non-Archimedean case and an operator of trace class in the real case of embedding θ : Z ′ ֒→ Z such that ν(Z ′ ) = 0. 13. Theorem. Let (G, τG ) and (H, τH ) be a pair of topological non-locally compact groups G, H (Banach-Lie, Frechet-Lie or groups of diffeomorphisms or loop groups) with uniformities τG , τH such that H is dense in (G, τG ) and there is a probability measure µ ∈ Ml (G, H) with continuous dµ (z, g) on H ×G. Also let X be a Hilbert space over C and U(X) be the unitary group. Then (1) if T : G → U(X) is a weakly continuous representation, then there exists T ′ : G → U(X) equal µ-a.e. to T and T ′ |(H,τH ) is strongly continuous; (2) if T : G → U(X) is a weakly measurable representation and X is separable, then there exists T ′ : G → U(X) equal to T µ-a.e. and T ′ |(H,τH ) is strongly continuous.

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Proof. Let R(G) := (I) ∪ L1 (G, µ), where I is the unit operator on L1 . Then we can define Z A(λe+a)h := λI + ah (g)[ρµ (h−1 , g)]Tg µ(dg), G

where ah (g) :=

a(h−1 g).

Then

|(A(λe+a)h − Aλe+a ξ, η)| ≤

Z

G

|ah (g)ρµ (h, g) − a(g)| |(Tg ξ, η)|µ(dg),

hence Aah is strongly continuous with respect to h ∈ H, that is, lim |Aah ξ − Aa ξ| = 0.

h→e

Denote Aah = Th′ Aa (see also § 29 [Nai68]), so Th′ ξ = Aah ξ, where ξ = Aa ξ0 , a ∈ L1 . Whence (Th′ ξ, Th′ ξ) = (Aah ξ0 , Aah ξ0 ) = Z

G

a¯h (g)(Tg ξ0 , Tg′ ξ0 )ρµ (h−1 , g)ρµ (h−1 , g′ )ah (g′ )µ(dg)µ(dg′ ) =

Z

G

′ a(z)a(z ¯ )(Uz ξ0 ,Uz′ ξ0 )µ(dz)µ(dz′ ) = (ξ, ξ).

Therefore, Th′ is uniquely extended to a unitary operator in the Hilbert space X ′ ⊂ X. In view of lemma 12, µ(H) = 0. Hence T ′ may be considered equal to T µ-a.e. Then a space spanC [Aah : h ∈ H] is evidently dense in X, since  Z Z ah (g)Tg ρµ (h−1 , g)µ(dg)ξ1 , xq (g′ )Tg′ ρµ (q−1 , g′ )µ(dg′ )ξ0 = (Aah ξ1 , Axq ξ0 ) = G

G

  Z Z ′ ′ = Th a(g)Tg µ(dg)ξ1 , Tq x(g )Tg′ µ(dg ) = (Tq−1 h Aa ξ1 , Ax ξ0 ). G

G

For proving the second statement let R := [ξ : Aa ξ = 0 for each a ∈ L1 (G, µ)]. If (Aa ξ, η) =

Z

G

a(g)(Tg ξ, η)µ(dg) =

Z

G

a(g)(Tg′ ξ, η)µ(dg)

for each a(g) ∈ L1 (G, µ, C), then (Tg ξ, η) = (Tg′ ξ, η) for µ-almost all g ∈ G. Suppose that {ξn : n ∈ N} is a complete orthonormal system in X. If ξ ∈ X, then Z

G

a(g)(Tg ξ, ξm )µ(dg) = 0

for each g ∈ G \ Sm , where µ(Sm ) = 0. Therefore, (Tg ξ, ξm ) = 0 for each m ∈ N, if S g ∈ G \ S, where S := ∞ m=1 Sm . Hence Tg ξ = 0 for each g ∈ G \ S, consequently, ξ = 0. Then (Tg ξn , ξm ) = (Tg′ ξn , ξm ) for each g ∈ G \ γn,m , where µ(γn,m ) = 0. Hence (Tg ξn , ξm ) = S (Tg′ ξn , ξm ) for each n, m ∈ N and each g ∈ G \ γ, where γ := n,m γn,m and µ(γ) = 0. Therefore, R = 0.

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14. Definition and note. Let {Gi : i ∈ No } be a sequence of topological groups such that G = G0 , Gi+1 ⊂ Gi and Gi+1 is dense in Gi for each i ∈ No and their topologies are denoted τi , τi |Gi+1 ⊂ τi+1 for each i, where No := {0, 1, 2, . . . }. Suppose that these groups are supplied with real probability quasi-invariant measures µi on Gi relative to Gi+1 . For example, such sequences exist for groups of diffeomorphisms or wrap (particularly loop) groups considered in previous papers [DS69, Sha89, Kos94, Lud96, Lud99t, Lud98s, Lud99r, Lud00a, Lud00d, Lud02b, Lud08]. 2 Let LG (Gi , µi , C) denotes a subspace of L2 (Gi , µi , C) as in § 1(b). Such spaces are Bai+1 nach and not Hilbert in general. Let L˜ 2 (Gi+1 , µi+1 , L2 (Gi , µi , C)) := Hi denotes the subspace of L2 (Gi , µi , C) of elements f such that 2

k f k2i := [k f k2L2 (Gi ,µi ,C) + k f k′ i ]/2 < ∞, where 2 k f k′ i

Z

:=

Z

| f (y−1 x)|2 µi (dx)µi+1 (dy).

Gi+1 Gi

Evidently Hi are Hilbert spaces due to the parallelogram identity. Let f

i+1

i

∗ f (x) :=

Z

f i+1 (y) f i (y−1 x)µi+1 (dy) Gi+1

denotes the convolution of f i ∈ Hi . 15. Lemma. The convolution ∗ : Hi+1 × Hi → Hi is the continuous bilinear mapping. Proof. In view of Fubini theorem and Cauchy inequality: Z

= Z

≤

Z Z

Gi Gi+1

Z

i

−1 −1

|f

i+1

2 i+1

2 i+1

(y)| µ

1/2 Z (dy)

k f i+1 k2L2 (Gi+1 ,µi+1 ,C) Z

f i+1 (y) f i (y−1 z−1 x)µi+1 (dy)

Gi+1 Gi Gi+1

Gi+1

| f (y z x)| µ

≤

Z Z

 ¯f i+1 (q) f¯i (q−1 z−1 x)µi+1 (dq)µi (dx)µi+1 (dz)

Gi+1

≤

| f i+1 ∗ f i (z−1 x)|2 µi (dx)µi+1 (dz)

Gi+1 Gi

Z

Gi+1

Z

Z

Z

Z Z Gi i

1/2  Z (dy)

|f

| f (q z x)| µ i

−1 −1

= k f i+1 k2L2 (Gi+1 ,µi+1 ,C)

1/2 µi (dx)µi+1 (dz) (dq)

2 i+1

| f (y z x)| µ

−1 −1

2 i+1

| f (q z x)| µ Z

(q)| µ

1/2 (dq)

i+1

(dy)µ

Gi+1 Gi+1 i+1

(dq)µ

Gi+1 Gi+1

k f i+1 k2L2 (Gi+1 ,µi+1 ,C)

2 i+1

Gi+1

Z

2 i+1

Gi+1

−1 −1

i

i+1

Z Z

1/2 (dz)

1/2 µi (dx) (dz)

| f i (y−1 z−1 x)|2 µi+1 (dy)µi+1 (dz)µi (dx)

Gi+1 Gi Gi+1

Z Z

Z

| f (y γ)| µ

Gi Gi+1 Gi+1

i

−1

2 i+1

i+1

(dy)µ

 (dz)dµi (z , γ)µ (dγ) −1

i

Algebras of Real Measures on Groups Z Z

≤ k f i+1 k2L2 (Gi+1 ,µi+1 ,C) since

Z Z

Gi Gi+1

i

| f i (z−1 x)|2 µi+1 (dz)µi (dx),

Gi Gi+1

dµi (z , γ)µ (dγ)µ −1

i+1

(dz) =

Z

i+1

µ

Z

(dz)

Gi+1

µi (zdγ) = 1. Gi

Then kf

i+1

∗

≤

f i k2L2 (Gi ,µi ,C)

=

Z Z Gi

k f i+1 k2L2 (Gi+1 ,µi+1 ,C)

151



f

i+1

i

−1

i+1

(y) f (y x)µ

Gi+1

Z Z

2 (dy) µi (dx)

| f i (z−1 x)|2 µi+1 (dz)µi (dx).

Gi Gi+1

Therefore, k f i+1 ∗ f i ki ≤ k f i+1 kL2 (Gi+1 ,µi+1 ,C) k f i ki . 16. Definition. Let l2 ({Hi : i ∈ No }) =: H be the Hilbert space consisting of elements f = ( f i : f i ∈ Hi , i ∈ No ), for which ∞

k f k2 := ∑ k f i k2i < ∞. i=0

For elements f and g ∈ H their convolution is defined by the formula: f ⋆ g := h with ∧ hi := f i+1 ∗gi for each i ∈ No . Let ∗ : H → H be an involution such that f ∗ := ( f¯ j : j ∈ No ), ∧ j where f j (y j ) := f j (y−1 j ) for each y j ∈ G j , f := ( f : j ∈ No ), z¯ denotes the complex conjugated z ∈ C. 17. Lemma. H is a non-associative non-commutative Hilbert algebra with involution ∗, that is ∗ is conjugate-linear and f ∗∗ = f for each f ∈ H. Proof. In view of Lemma 15 the convolution h = f ⋆ g in the Hilbert space H has the norm khk ≤ k f k kgk, hence is a continuous mapping from H × H into H. From its definition it follows that the convolution is bilinear. It is non-associative as follows from the computation of i-th terms of ( f ⋆ g) ⋆ q and f ⋆ (g ⋆ q), which are ( f i+2 ∗ gi+1 ) ∗ qi and f i+1 ∗ (gi+1 ∗ qi ) respectively, where f , g and q ∈ H. It is non-commutative, since there are ∧∧ f and g ∈ H for which f i+1 ∗ gi are not equal to gi+1 ∗ f i . Since f j (y j ) = f j (y j ) and z¯ = z, one has f ∗∗ = ( f ∗ )∗ = f . 18. Note. In general ( f ⋆ g∗ )∗ 6= g ⋆ f ∗ for f and g ∈ H, since there exist f j and g j such that g j+1 ∗ ( f j )∗ 6= ( f j+1 ∗ (g j )∗ )∗ . If f ∈ H is such that f j |G j+1 = f j+1 , then (( f j+1 )∗ ∗ f j )(e) =

Z

G j+1

f¯ j+1 (y−1 ) f j+1 (y)µ j+1 (dy) = k f j+1 k2L2 (G j+1 ,µ j+1 ,C) ,

where j ∈ No . 19. Definition. Let l2 (C) the standard Hilbert space over the field C be considered as a Hilbert algebra with the convolution α ⋆ β = γ such that γi := αi+1 βi , where α := (αi : αi ∈ C, i ∈ No ), α, β and γ ∈ l2 (C). 20. Note. The algebra l2 (C) has two-sided ideals Ji := {α ∈ l2 (C) : α j = 0 for each j > i}, where i ∈ No . That is, J ⋆ l2 (C) ⊂ J and l2 (C) ⋆ J = J and J is the C-linear subspace of l2 (C), but J ⋆ l2 (C) 6= J. There are also right ideals, which are not left ideals: Ki := {α ∈ l2 (C) : α j = 0 for each j = 0, . . . , i}, where j ∈ No . That is, l2 (C) ⋆ Ki = Ki , but

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Ki ⋆ l2 (C) = Ki−1 for each i ∈ No , where K−1 := l2 (C). The algebra l2 (C) is the particular case of H, when G j = {e} for each j ∈ No . We consider further H for non-trivial topological groups outlined above. 21. Theorem. If F is a maximal proper left or right ideal in H, then H/F is isomorphic as the non-associative noncommutative algebra over C with l2 (C). Proof. Since F is the ideal, it is the C-linear subspace of H. Suppose, that there exists j ∈ No such that f j = 0 for each f ∈ F, then f i = 0 for each i ∈ No , since the space of T bounded complex-valued continuous functions Cb0 (G∞ , C) on G∞ := ∞j=0 G j is dense in each H j := { f j : f ∈ H} and Cb0 (G∞ , C) ∩ Fj = {0} and Cb0 (G j , C)|G j+1 ⊃ Cb0 (G j+1 , C). Therefore, Fj 6= {0} for each j ∈ No , consequently, C ֒→ Fj for each j ∈ No . Since C is embeddable into each Fj , then there exists the embedding of l2 (C) into F, where H j := { f j : f ∈ H}, π j : H → H j are the natural projections. The subalgebra F is closed in H, since H is the topological algebra and F is the maximal T proper subalgebra. The space H∞ := j∈No H j is dense in each H j and the group G∞ := T j∈No G j is dense in each G j . Suppose that Fi = Hi for some i ∈ No , then Fj = H j for each j ∈ No , since Cb0 (G∞ , C) is dense in each H j and Cb0 (G j , C)|G j+1 ⊃ Cb0 (G j+1 , C). The ideal F is proper, consequently, Fj 6= H j as the C-linear subspace for each j ∈ No , where Fj = π j (F). There are linear continuous operators from l2 (C) into l2 (C) given by the following formulas: x 7→ (0, . . . , 0, x0 , x1 , x2 , . . . ) with 0 as n coordinates at the beginning, x 7→ (xn , xn+1 , xn+2 , . . . ) for n ∈ N; x 7→ (xkl+σk (i) : k ∈ No , i ∈ (0, 1, . . . , l − 1)), where N ∋ l ≥ 2, σk ∈ Sl are elements of the symmetric group Sl of the set (0, 1, . . . , l − 1). Then f ⋆ (g ⋆ h) + l2 (C) and ( f ⋆ g) ⋆ h + l2 (C) are considered as the same class, also f ⋆ g + l2 (C) = g ⋆ f + l2 (C) in H/l2 (C), since ( f + l2 (C)) ⋆ (g + l2 (C)) = f ⋆ g + l2 (C) for each f , g and h ∈ H. For each f , g, h ∈ F: f ⋆ (g ⋆ h) + l2 (C) and ( f ⋆ g) ⋆ h + l2 (C) are considered as the same class, also f ⋆ g + l2 (C) = g ⋆ f + l2 (C) in F/l2 (C), since ( f + l2 (C)) ⋆ (g + l2 (C)) = f ⋆ g + l2 (C) ⊂ F for each f and g ∈ F. Therefore, the quotient algebras H/l2 (C) and F/l2 (C) are the associative commutative Banach algebras. Let us adjoint a unit to H/l2 (C) and F/l2 (C). As a consequence of the Gelfand and Mazur theorem we have, that (H/l2 (C))/(F/l2 (C)) is isomorphic with C (see also Theorem V.6.12 [FD88] and Theorem III.11.1 [Nai68]). On the other hand, as it was proved above Fj 6= H j for each j ∈ No , hence there exists the following embedding l2 (C) ֒→ (H/F) and (H/F)/l2 (C) is isomorphic with (H/l2 (C))/(F/l2 (C)). Therefore, H/F is isomorphic with l2 (C).

3.3.

Comments

Another methods of construction of unitary representations of topological totally disconnected groups which may be non-locally compact with the help of quasi-invariant realvalued measures were given in [Lud98b, Lud02b, Lud00a, Lud99t, Lud01s, Lud0348, Lud01f, Lud08, LD03] and references therein.

Chapter 4

Algebras of Non-Archimedean Measures on Groups 4.1.

Introduction

In Chapter II quasi-invariant measures on Banach spaces with values in fields supplied with non-Archimedean multiplicative norms were studied. Besides Banach spaces they were constructed and investigated on non-locally compact topological groups. Quasiinvariant measures with values in non-Archimedean fields on a group of diffeomorphisms were constructed for non-Archimedean manifolds M in [Lud96, Lud99t, Lud08]. On non-Archimedean wrap (particularly loop) groups and semigroups they were provided in [Lud98s, Lud00a, Lud02b, Lud08]. A Banach space over a locally compact field also serves as the additive group and quasi-invariant measures on it were studied in Chapter 2. This chapter is devoted to the investigation of properties of quasi-invariant measures with values in non-Archimedean fields that are important for analysis on topological groups and for construction of irreducible representations. The following properties are investigated: (1) convolutions of measures and functions, (2) continuity of functions of measures, (3) non-associative algebras generated with the help of quasi-invariant measures. The theorems given below show that many differences appear to be between locally compact and non-locally compact groups. Algebras of measures and functions on groups are considered below. The groups considered below are supposed to have structure of Banach manifolds over the corresponding fields if something other is not specified.

4.2.

Algebras of Measures and Functions

1. Definitions. (a). Let G be a Hausdorff separable topological group. A tight measure µ on A f (G, µ) with values in a non-Archimedean field F is called left-quasi-invariant (or right) relative to a dense subgroup H of G, if µφ (∗) (or µφ (∗)) is equivalent to µ(∗) for each φ ∈ H, where Bco(G) is the algebra of all clopen subsets of G, A f (G, µ) denotes its completion by µ, µφ (A) := µ(φ−1 A), µφ (A) := µ(Aφ−1 ) for each A ∈ A f (G, µ), ρµ (φ, g) :=

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µφ (dg)/µ(dg) ∈ L(G, A f (G, µ), µ, F) (or ρ˜ µ (φ, g) := µφ (dg)/µ(dg)) denotes a left (or right) quasi-invariance factor, F is a non-Archimedean field complete relative to its uniformity and such that Ks ⊂ F. We assume that a uniformity τG on G is such that τG |H ⊂ τH , (G, τG ) and (H, τH ) are complete. We suppose also that there exists an open base in e ∈ H such that their closures in G are compact. Such pairs exist for wrap (particularly loop) groups and groups of diffeomorphisms and Banach-Lie groups. We denote by Ml (G, H) (or Mr (G, H)) a set of left-( or right) quasi-invariant tight measures on G relative to H with a finite norm kµk < ∞. (b). Let LH (G, µ, F) denotes the Banach space of functions f : G → F such that fh (g) ∈ L(G, µ, F) for each h ∈ H and k f kLH (G,µ,F) := sup k fh kL(G,µ,F) < ∞, h∈H

where F is a non-Archimedean field for which Ks ⊂ F, fh (g) := f (h−1 g) for each g ∈ G. For µ ∈ Ml (G, H) and ν ∈ M(H) let (ν ∗ µ)(A) :=

Z

H

µh (A)ν(dh) and (q˜∗ f )(g) :=

Z

f (hg)q(h)ν(dh) H

be convolutions of measures and functions, where M(H) is the space of tight measures on H with a finite norm, ν ∈ M(H) and q ∈ L(H, ν, F). 2. Lemma. The convolutions are continuous F-linear mappings ∗ : M(H) × Ml (G, H) → Ml (G, H) and ∗˜ : L(H, ν, F) × LH (G, µ, F) → LH (G, µ, F). Proof. If X and Y are sets with separating covering rings R and S and measures µ and ν on them, then the Banach space L(µ × ν) is linearly topologically isomorphic with ˆ the (Banach completed) tensor product L(µ)⊗L(ν) of Banach spaces L(µ) and L(ν) (see Theorem 7.16 and Chapter 4 in [Roo78]). By the definition this means that if E and F are ˆ is their (Banach completed) tensor product two Banach spaces over the same field and E ⊗F ˆ is characterized by two conditions: then the mapping E × F ∋ (x, y) 7→ (x ⊗ y) ∈ E ⊗F (Bi) kx ⊗ yk ≤ kxkkyk for all x ∈ X and y ∈ Y ; (Bii) for each continuous bilinear mapping S of E × F into any Banach space H over the same field there exists a unique continuous linear bounded operator S⊗ with the norm kS⊗ k ≤ kSk and such that S⊗ ◦ θ(x, y) = S(x, y) ∈ H for all x ∈ E and y ∈ F, where θ : ˆ is the natural embedding. E × F ֒→ E ⊗F In view of the aforementioned theorem and estimates kν ∗ µk ≤ kνk × kµk, kq˜∗ f kLH (G,µ,F) ≤ kqkL(H,ν,F) × k f kLH (G,µ,F) , since ρµ (h, g) ∈ L(H × G, ν × µ, F), we get the statement of this lemma. 3. Lemma. For µ ∈ Ml (G, H) the translation map (q, f ) → fq (g) is continuous from H × LH (G, µ, F) into LH (G, µ, F). Proof. In view of Lemma 7.10 and Theorem 7.12 [Roo78] recalled in § 2.32 above for each ε > 0 the set {x : | f (x)|Nµ (x) ≥ ε} is A f (G, µ)-compact and f is A f (G, µ)-continuous.

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The embedding of H into G is compact (see § 1), hence for each q ∈ H there exists V clopen in H and such that q−1V is a subgroup of H with clG q−1V compact in G, where clG (A) denotes the closure of a subset A in G. The product of compact subsets in G is compact in G, hence fq (g) ∈ LH (G, µ, F) and (q, f ) 7→ fq (g) is the continuous mapping, since the restriction of the Bco(G) and the A f (G, µ)-topologies onto Xε coincide, k fq kLH (µ) = k f kLH (µ) for each q ∈ H (see § 1.(b)). 4. Proposition. For a probability measure µ ∈ M(G) there exists anRapproximate unit which is a sequence of nonzero continuous functions ψi : G → F such that G ψi (g)µ(dg) = 1 and for each neighborhood U ∋ e in G there exists i0 such that supp(ψi ) ⊂ U for each i > i0 . Proof. A group G has a countable base of neighborhoods of e ∈ G. A measure µ is quasi-invariant, hence kUkµ > 0 for each neighborhood U ∋ e, µ is the tight measure, hence T there exists a system of neighborhoods {Ui : Ui ∋ e∀i}, i Ui = {e}, Ui ⊃ Ui+1 for each i, R supp(ψi ) ⊂ Ui . Choose ψi such that G ψi (g)µ(dg) = 1 for each i. 5. Proposition. If (ψi : i ∈ N) is an approximate unit in H relative to a probability measure ν ∈ M(H), then limi→∞ ψi ∗ f = f in the LH (G, µ, F) norm, where µ ∈ Ml (G, H), f ∈ LH (G, µ, F). Proof. In view of Theorem 7.12 [Roo78] recalled in § 2.32 above for each ε > 0 and each i there exists a finite number of h j ∈ H, j = 1, . . . , n, n ∈ N, such that Sn j=1 h jUi ⊃ Xε, f × V, where {x : | f (x)|Nµ (x) ≥ ε} =: Xε, f , V is a clopen neighborhood of ξ ∈ H which can be chosen such that ξ−1V is a subgroup of H, clG (ξ−1V ) is comR pact in G, (ψi ∗ f )(g) = h∈Ui ψi (h) f (hg)ν(dh) and supg∈G | fξ (g) − (ψi ∗ fξ )(g)|Nµ (g) ≤ supg∈G [suph∈Xε, f |ψi (h)|| fξ (hg) − fξ (g)|Nµ (g) + εkψi kν kGkµ ], hence limi→∞ (ψi ∗ f ) = f in ξ

LH (G, µ, F)-norm. 6. Lemma. Suppose g ∈ LH (G, µ, F) and (gx |H ) ∈ L(H, ν, F) for each x ∈ G, f ∈ L(H, ν, F), where gx (y) := g(yx) for each x and y ∈ G. Let µ and ν be probability measures, µ ∈ Ml (G, H), ν ∈ M(H). Then f ∗˜ g ∈ LH (G, µ, F) and there exists a function h : G → F such that h|H is continuous, h = f ∗˜ g µ-a.e. on G and h vanishes at ∞ on G. Proof. In view of the Fubini theorem we have Z f (y)g(yz)ν(dy) Nµ (z) k f ∗˜ gkLH (G,µ,F) = sup h∈H,z∈G

H

≤ kg(z)kLH (G,µ,F) k f kL(H,ν,F)

The equation α f (φ) :=

Z

f (y)φ(y)ν(dy) H

defines a continuous linear functional on the Banach space L∞ (H, ν, F) := {φ : H → F : φ is (A f (H, ν), Bco(F))-measurable, kφk∞ := essν − sup |φ(x)| < ∞}. x∈H

In view of Lemma 3 the function −1

α f (g(qx) ) =: v(qx) =: w(q, x)

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of two variables q and x is continuous on H × H for q, x ∈ H, since the mapping (q, x) 7→ (qx)−1 is continuous from H × H into H. By Theorem 7.16 [Roo78] recalled in § 2 Z

v(y)ψ(y)µ(dy) =

Z Z

f (y)g(yx)ψ(x)ν(dy)µ(dx)

G H

G

= L∞ (G, µ, F).

Z

H

 Z g(yx)ψ(x)µ(dx) ν(dy) f (y) G

for each ψ ∈ From this it follows, that µ({y : h(y) 6= ( f ∗˜ g)(y), y ∈ G}) = 0, since h and ( f ∗˜ g) are µ-measurable functions due to Fubini theorem and the continuity of the composition and the inversion in a topological group. In view of Theorem 7.12 [Roo78] recalled in § 2.32 for each ε > 0 there are compact subsets C ⊂ H and D ⊂ G and functions f ′ ∈ L(H, ν, F) and g′ ∈ LH (G, µ, F) with closed supports supp( f ′ ) ⊂ C, supp(g′ ) ⊂ D such that clGCD is compact in G, k f ′ − f kL(H,ν,F) < ε and kg′ − gkLH (G,µ,F) < ε, since by the supposition of § 1 the group H has the base BH of its topology τH , such that the closures clGV are compact in G for each V ∈ BH . From the inequality |h′ (x) − h(x)| ≤ (k f kL(H,ν,F) + ε)ε + εkgkLH (G,µ,F) it follows that for each δ > 0 there exists a compact subset K ⊂ G with |h(x)| < δ for each x ∈ G \ K, where h′ (x−1 ) := α f ′ (g′ x ). 7. Proposition. Let A, B ∈ A f (G, µ), µ and ν be probability measures, µ ∈ Ml (G, H), ν ∈ M(H). Then the function ζ(x) := µ(A ∩ xB) is continuous on H and ν(yB−1 ∩ H) ∈ L(H, ν, F). Moreover, if kAkµ kBkµ > 0, µ({y ∈ G : yB−1 ∩ H ∈ A f (H, ν) and kyB−1 ∩ Hkν > 0}) > 0, then ζ(x) 6= 0 on H. Proof. Let gx (y) := ChA (y)ChB (x−1 y), then gx (y) ∈ LH (G, µ, F), where ChA (y) is the characteristic function of A. In view of Propositions 4 and 5 there exists limi→∞ ψi ∗ gx = gx in LH (G, µ, F). In view of Lemma 6 ζ(x)|H is continuous. There is the following inequality: Z Z Z −1 1 ≥ µ(A ∩ xB)ν(dx) = ChA (y)ChB (x y)µ(dy)ν(dx) . H

H G

In view of Theorem 7.16 [Roo78] recalled in § 2 there exists Z

H

ChB (x−1 y)ν(dy) = ν((yB−1 ) ∩ H) ∈ L(G, µ, F), hence Z

µ(A ∩ xB)ν(dx) = H

Z

G

ν(yB−1 ∩ H)ChA (y)µ(dy).

8. Corollary. Let A, B ∈ A f (G, µ), ν ∈ M(H) and µ ∈ Ml (G, H) be probability measures. Then denoting IntH V the interior of a subset V of H with respect to τH , one has / when (i) IntH (AB) ∩ H 6= 0, k{y ∈ G : kyB ∩ Hkν > 0}kµ > 0;

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(ii) IntH (AA−1 ) ∋ e, when k{y ∈ G : kyA−1 ∩ Hkν > 0}kµ > 0. Proof. We infer the inclusion AB ∩ H ⊃ {x ∈ H : k(A ∩ xB−1 )kµ > 0} demonstrating this corollary. 9. Corollary. Let G = H. If µ ∈ Ml (G, H) is a probability measure, then G is a locally compact topological group. Proof. Let us take ν = µ and A = C ∪ C−1 , where C is a compact subset of G with k(C)kµ > 0, whence k(yA)kµ > 0 for each y ∈ G and inevitably IntG (AA−1 ) ∋ e. 10. Corollary. Let µ ∈ Ml (G, H), ρµ (h, z) ∈ L(H, ν, F) × L(G, µ, F), then ρµ (h, z) is continuous ν × µ-a.e. on H × G. Proof. In view of the co-cycle condition ρµ (φψ, g) := µφψ (dg)/µ(dg) = (µφψ (dg)/µφ (dg))(µφ (dg)/µ(dg)) = ρµ (ψ, φ−1 g)ρµ (φ, g) on ρµ and Corollary 8 above, Theorem 7.12 [Roo78] recalled in § 2.32 for each ε > 0 the quasi-invariance factor ρµ (h, g) is continuous on H × Gε , but Gε is neighborhood of e in G, where Gε := {g ∈ G : Nµ (g) ≥ ε}. Since H is dense in G and G is separable, S then ∞j=1 h j Gε = G, where {h j : j ∈ N} is a countable subset in G. Therefore, ρµ (h, z) is continuous ν × µ-a.e. on H × G. 11. Corollary. Let G be a locally compact group, ρµ (h, z) ∈ L(G × G, µ × µ, F), then ρµ (h, z) is µ × µ-a.e. continuous on G × G. Proof. It follows from Corollaries 9 and 10. 12. Remark. The latter two corollaries show, that the condition of continuity of ρµ (h, z) imposed in Chapter 2 is not restrictive. 13. Lemma. Let µ ∈ Ml (G, H) be a probability measure and G be non-locally compact. Then kHkµ = 0. Proof. This follows from Theorem 2.3.13 above and the proof of Lemma 2, since the embedding Te H ֒→ Te G is a compact operator in the non-Archimedean case and a tight measure µ on G induces a tight measure on a neighborhood V of 0 in Te G such that V is topologically homeomorphic to a clopen subgroup U in G (see also Chapter 3 and the papers about construction of quasi-invariant measures on the considered here groups [Lud96, Lud99t, Lud98s, Lud00a, Lud02b]). 14. Let (G, τG ) and (H, τH ) be a pair of topological non-locally compact groups G, H (Banach-Lie, Frechet-Lie or groups of diffeomorphisms or loop groups) with uniformities τG , τH such that H is dense in (G, τG ) and there is a probability measure µ ∈ Ml (G, H) with continuous ρµ (z, g) on H × G. Also let X be a Banach space over F and IS(X) be the group of isometric F-linear automorphisms of X in the topology inherited from the Banach space L(X) of all bounded F linear operators from X into X. Theorem. (1). If T : G → IS(X) is a weakly continuous representation, then there exists T ′ : G → IS(X) equal µ-a.e. to T and T ′ |(H,τH ) is strongly continuous. (2). If T : G → IS(X) is a weakly measurable representation and X is of separable type c0 (F) over F, then there exists T ′ : G → IS(X) equal to T µ-a.e. and T ′ |(H,τH ) is strongly continuous.

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Proof. Take K (G) := (I) ∪ L(G, µ, F), where I is the unit operator on L(G, µ, F). Then we can define Z A(λe+a)h := λI + ah (g)[ρµ (h, g)]Tg µ(dg), G

where ah (g) :=

a(h−1 g).

Then

|(A(λe+a)h − Aλe+a ξ, η)| ≤ sup[|ah (g)ρµ (h, g) − a(g)| |Tg ξ|Nµ (g)], g∈G

hence Aah is strongly continuous with respect to h ∈ H, that is, lim |Aah ξ − Aa ξ| = 0.

h→e

Denote Aah = Th′ Aa , so Th′ ξ = Aah ξ0 , where ξ = Aa ξ0 , a ∈ L(G, µ, F). Whence T ′ h ξ = A ah ξ 0 = = Th

Z

G

Z

G

ah (g)Tg ξ0 ρµ (h, g)µ(dg)

a(z)Tz ξ0 µ(dz) = Th ξ,

hence |T ′ h ξ| = |ξ| for each h ∈ H. Therefore, Th′ is uniquely extended to an isometric operator on the Banach space X ′ ⊂ X. In view of Lemma 10, kHkµ = 0. Hence T ′ may be considered equal to T µ-a.e. Then a space spanF [Aah : h ∈ H] is evidently dense in X, since A ah ξ =

Z

G

= Th

ah (g)Tg ρµ (h, g)µ(dg)ξ Z

G

a(g)Tg µ(dg)ξ.

For proving the second statement let R := [ξ : Aa ξ = 0 for each a ∈ L(G, µ, F)]. If η ∈ X ∗ and   Z Z ′ a(g)T g ξµ(dg) a(g)Tg ξµ(dg) = η η(Aa ξ) = η G

G

for each a(g) ∈ L(G, µ, F), then η(Tg ξ) = η(Tg′ ξ) for µ-almost all g ∈ G, where X ∗ denotes the topological dual space of all K-linear functionals f : X → K. Suppose that {ξ∗n : n ∈ N} is an orthonormal system in X ∗ separating points of X. It exists, since by the supposition of this theorem X = c0 (F). If ξ ∈ X, then  Z ∗ a(g)Tg ξµ(dg) = 0 ξm G

for each g ∈ G \ Sm , where kSm kµ = 0. Therefore, ξ∗m (Tg ξ) = 0 for each m ∈ N, if g ∈ G \ S, S where S := ∞ m=1 Sm . Hence Tg ξ = 0 for each g ∈ G \ S, consequently, ξ = 0. Consider the embedding X ֒→ X ∗ with the help of the standard orthonormal basis {e j : j} in X over F. Therefore, let {ξm : m ∈ N} ⊂ X ֒→ X ∗ . Then ξ∗m (Tg ξn ) = ξ∗m (Tg′ ξn ) for each g ∈ G \ γn,m , where kγn,m kµ = 0, ξ∗m is the image of ξm under this standard embedding X ֒→ X ∗ . Hence

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ξ∗m (Tg ξn ) = ξ∗m (Tg′ ξn ) for each n, m ∈ N and each g ∈ G \ γ, where γ := n,m γn,m and kγkµ = 0, since in L(G, µ, F) the family of all step functions is dense. Therefore, we get R = 0. 15. Definition and note. Let {Gi : i ∈ No } be a sequence of topological groups such that G = G0 , Gi+1 ⊂ Gi and Gi+1 is dense in Gi for each i ∈ No and their topologies are denoted τi , τi |Gi+1 ⊂ τi+1 for each i, where No := {0, 1, 2, . . . }. Suppose that these groups are supplied with F-valued probability quasi-invariant measures µi on Gi relative to Gi+1 . For example, such sequences exist for groups of diffeomorphisms or wrap (particularly loop) groups considered in previous papers [Lud96, Lud99t, Lud98s, Lud00a, Lud02b, Lud08]). Let LGi+1 (Gi , µi , F) denotes the Banach subspace of L(Gi , µi , F) as in § 1(b). Let ˜ i+1 , µi+1 , L(Gi , µi , F)) =: Hi denotes the completion of the subspace of L(Gi , µi , F) of L(G all elements f such that S

k f ki := max[k f 2 k1/2 L(Gi ,µi ,F) , k f k′ i ] < ∞, where k f k′ i := [

sup x∈Gi ,y∈Gi+1

| f (y−1 x)|2 Nµi (x) max(1, Nµi+1 (y))]1/2 .

Evidently Hi are Banach spaces over F. Let f

i+1

i

∗ f (x) :=

Z

f i+1 (y) f i (y−1 x)µi+1 (dy) Gi+1

denotes the convolution of f i ∈ Hi . 16. Lemma. The convolution ∗ : Hi+1 × Hi → Hi is the continuous F-bilinear mapping. Proof. From the definitions we have: 1/2

1/2

k( f i+1 ∗ f i )2 kL(Gi ,µi ,F) = sup |( f i+1 ∗ f i )(x)|Nµi (x) x∈Gi

≤

1/2

sup x∈Gi ,y∈Gi+1

1/2

1/2

[| f i+1 (y)|Nµi+1 (y)][| f i (y−1 x)|Nµi (x)Nµi+1 (y)] 1/2

= k( f i+1 )2 kL(Gi+1 ,µi+1 ,F) k f i k′ i and k f i+1 ∗ f i k′ i = [ ≤[

sup

x∈Gi ,y∈Gi+1 ,z∈Gi+1

sup x∈Gi ,y∈Gi+1

|( f i+1 ∗ f i )(y−1 x)|2 Nµi (x)Nµi+1 (y)]1/2

| f i+1 (z)|2 Nµi+1 (z)| f i (y−1 z−1 x)|2 Nµi (x)Nµi+1 (y)Nµi+1 (z)]1/2 1/2

≤ k( f i+1 )2 kL(Gi+1 ,µi+1 ,F) k f i k′ i , since from A ∋ y and B ∋ z for A, B ∈ A f (Gi+1 , µi+1 ) it follows that AB ∋ yz and AB ∈ A f (Gi+1 , µi+1 ), which follows from i+1

µ

(AB) =

Z

i+1

µ A∋a

i+1

(aB)µ

(da) =

Z

Z

µi+1 (adb)µi+1 (da),

B∋b A∋a

so that kAkµi+1 ≤ kGi+1 kµi+1 = 1, hence Nµi+1 (z) ≤ 1 for each z ∈ Gi+1 for the probability measure µi+1 . Therefore, k f i+1 ∗ f i ki ≤ k f i+1 ki+1 k f i ki and inevitably the convolution ∗ : Hi+1 × Hi → Hi is the continuous F-bilinear mapping.

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17. Definition. Let c0 ({Hi : i ∈ No }) =: H be the Banach space consisting of elements f = ( f i : f i ∈ Hi , i ∈ No ), for which limi→∞ k f i ki = 0, where ∞

k f k := sup k f i ki < ∞. i=0

For elements f and g ∈ H their convolution is defined by the formula: f ⋆ g := h with ∧ hi := f i+1 ∗gi for each i ∈ No . Let ∗ : H → H be an involution such that f ∗ := ( f j : j ∈ No ), ∧ j where f j (y j ) := f j (y−1 j ) for each y j ∈ G j , f := ( f : j ∈ No ). 18. Lemma. H is a non-associative non-commutative Banach algebra with involution ∗, that is ∗ is F-bilinear and f ∗∗ = f for each f ∈ H. Proof. In view of Lemma 16 the convolution h = f ⋆ g in the Banach space H has the norm khk ≤ k f k kgk, hence is a continuous mapping from H × H into H. From its definition it follows that the convolution is F-bilinear. It is non-associative as follows from the computation of i-th terms of ( f ⋆ g) ⋆ q and f ⋆ (g ⋆ q), which are ( f i+2 ∗ gi+1 ) ∗ qi and f i+1 ∗ (gi+1 ∗ qi ) respectively, where f , g and q ∈ H. It is non-commutative, since there are ∧∧ f and g ∈ H for which f i+1 ∗ gi are not equal to gi+1 ∗ f i . From f j (y j ) = f j (y j ) it follows that f ∗∗ = ( f ∗ )∗ = f . 19. Note. In general ( f ⋆ g∗ )∗ 6= g ⋆ f ∗ for f and g ∈ H, since there exist f j and g j such that g j+1 ∗ ( f j )∗ 6= ( f j+1 ∗ (g j )∗ )∗ . If f ∈ H is such that f j |G j+1 = f j+1 , then (( f

j+1 ∗

j

) ∗ f )(e) =

Z

( f j+1 (y))2 µ j+1 (dy), hence G j+1 1/2

|(( f j+1 )∗ ∗ f j )(e)| ≤ k( f j+1 )2 kL(G j+1 ,µ j+1 ,F) ≤ k f j+1 k j+1 , where j ∈ No . 20. Definition. Consider the standard Banach space c0 (F) over the field F as a Banach algebra with the convolution α⋆β = γ such that γi := αi+1 βi , where α := (αi : αi ∈ F, i ∈ No ), α, β and γ ∈ c0 (F). 21. Note. The algebra c0 (F) has two-sided ideals Ji := {α ∈ c0 (F) : α j = 0 for each j > i}, where i ∈ No . That is, J ⋆ c0 (F) ⊂ J and c0 (F) ⋆ J = J and J is the F-linear subspace of c0 (F), but J ⋆ c0 (F) 6= J. There are also right ideals, which are not left ideals: Ki := {α ∈ c0 (F) : α j = 0 for each j = 0, . . . , i}, where j ∈ No . That is, c0 (F) ⋆ Ki = Ki , but Ki ⋆ c0 (F) = Ki−1 for each i ∈ No , where K−1 := c0 (F). The algebra c0 (F) is the particular case of H, when G j = {e} for each j ∈ No . We consider further H for non-trivial topological T groups outlined above with G∞ := ∞j=0 G j dense in each G j . 22. Theorem. If F is a maximal proper left or right ideal in H, then H/F is isomorphic as the non-associative noncommutative algebra over F with c0 (F). Proof. The ideal F is also the F-linear subspace of H. In view of Theorem 7.12 [Roo78] recalled above in § 2.32 a function f : G j → F is µ j -integrable if and only if it satisfies two properties: f is A f (G, µ j )-continuous and for each ε > 0 the set {x : | f (x)|Nµ j (x) ≥ ε} is A f (G, µ j )-compact and hence contained in {x : Nµ j (x) ≥ δ} for some δ > 0. Suppose, that there exists j ∈ No such that f j = 0 for each f ∈ F , then f i = 0 for each i ∈ No , since the space of bounded F-valued continuous functions Cb0 (G∞ , F) on G∞ is dense H j := { f j : f ∈ H} and Cb0 (G∞ , F) ∩ Fj = {0} and Cb0 (G j , F)|G j+1 ⊃ Cb0 (G j+1 , F).

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Therefore, F j 6= {0} for each j ∈ No , where F j := { f j : f ∈ F }, consequently, F ֒→ Fj for each j ∈ No . Since F is embeddable into each Fj , then there exists the embedding of c0 (F) into F , where H j := { f j : f ∈ H}, π j : H → H j are the natural projections. The subalgebra F is closed in H, since H is the topological algebra and F is the maxiT mal proper subalgebra. The space H∞ := j∈No H j is dense in each H j . If F i = Hi for some i ∈ No , then F j = H j for each j ∈ No , since Cb0 (G∞ , F) is dense in each H j and Cb0 (G j , F)|G j+1 ⊃ Cb0 (G j+1 , F). The ideal F is proper, consequently, F j 6= H j as the F-linear subspace for each j ∈ No , where F j = π j (F ). There exist F-linear continuous operators from c0 (F) into c0 (F) such that x 7→ (0, . . . , 0, x0 , x1 , x2 , . . . ) with 0 as n coordinates at the beginning, x 7→ (xn , xn+1 , xn+2 , . . . ) for n ∈ N; x 7→ (xkl+σk (i) : k ∈ No , i ∈ (0, 1, . . . , l − 1)), where N ∋ l ≥ 2, σk ∈ Sl are elements of the symmetric group Sl of the set (0, 1, . . . , l − 1). Then f ⋆ (g ⋆ h) + c0 (F) and ( f ⋆ g) ⋆ h + c0 (F) are considered as the same class, also f ⋆ g + c0 (F) = g ⋆ f + c0 (F) in H/c0 (F), since ( f + c0 (F)) ⋆ (g + c0 (F)) = f ⋆ g + c0 (F) for each f , g and h ∈ H. Then f ⋆ (g ⋆ h) + c0 (F) and ( f ⋆ g) ⋆ h + c0 (F) are considered as the same class for each f , g, h ∈ F , also f ⋆ g + c0 (F) = g ⋆ f + c0 (F) in F /c0 (F), since ( f + c0 (F)) ⋆ (g + c0 (F)) = f ⋆ g + l2 (F) ⊂ F for each f and g ∈ F . Therefore, the quotient algebras H/c0 (F) and F /c0 (F) are the associative commutative Banach algebras. From µi ∈ Ml (Gi , Gi+1 ) it follows that for each open subset W ∋ e, W ⊂ Gi there exists a clopen subgroup U ⊂ W such that µi (U) 6= 0, since otherwise µi (zV ) = 0 for each z ∈ Gi+1 and each open V ⊂ W , hence µi (Gi ) = 0 contradicting supposition, that each µi is the probability measure. Let us adjoin a unit to H/c0 (F) and to F /c0 (F). There is satisfied the equality ChGi+1 ∗ ChGi = ChGi . Let Ui, j be a clopen subgroup in Gi , that is possible, since each Gi is ultrametrizable. Choose Ui, j such that Ui, j+1 ⊂ Ui, j ∩ G j+1 and Ui, j ⊃ Ui+1, j T for each i and j, i Ui, j = e ∈ G j for each j. Since µ j (G j ) = 1 and kG j kµ j = 1, then by induction (i, j) ∈ {(1, 1), (1, 2), . . . , (1, n), . . . ; (2, 1), (2, 2), . . . , (2, n), . . . ; . . . , (m, 1), (m, 2), . . ., (m, n), . . . }, where m, n ∈ N, there exists a family αi, j ∈ F and {Ui, j : i, j} such that αi, j+1ChUi, j+1 ∗ αi, jChUi, j = αi, jChUi, j and 0 < |αi, j ||µi (Ui, j )| ≤ 1 for each i, j. Put ei := {αi, jChUi, j : j ∈ No }, then ei ∗ ei = ei for each i. From the properties of Ui, j it follows, that spanF {ei (z−1 g) : i ∈ N, z ∈ G∞ } is dense in H, where g := (g j : g j ∈ G j ∀ j ∈ No ), z−1 g = (z−1 g j : j). Consider the algebras H/c0 (F) =: A and F /c0 (F) =: B. The algebras A and B are commutative and associative. From the preceding proof it follows that spanF {ei (z−1 g) + c0 (F) : i ∈ N, z ∈ G∞ } is dense in A, each ei (z−1 g) + c0 (F) is the idempotent element in A. Remind Van der Put’s theorem about C-algebras. The following conditions on a commutative Banach algebra A over a field K are equivalent: (α) A is a C-algebra; (β) A+ is a C-algebra; (γ) the linear span of {e ∈ A : e = e2 , kek ≤ 1} is dense in A; (δ) every K[a] with a ∈ A is a C-algebra; (ε) A is the smallest closed subalgebra of A that contains {a ∈ A : K[a] is a C-algebra}, where if A does not contain a unit element A+ denotes the Banach algebra obtained from A by adding the unit element as K ⊕ A (see also Theorem 6.12 [Roo78]). Remind that a commutative Banach algebra A is called a C-algebra if there exists a locally

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compact zero-dimensional Hausdorff space X such that A is isomorphic with C∞ (X ). A normed algebra is an algebra A with a norm such that kxyk ≤ kxkkyk for all x, y ∈ A. A Banach algebra is a complete normed algebra. As usually K[x] denotes the ring of polynomials over a field K, clX B = B¯ denotes the closure of a subset B in a topological space X. Therefore, by the aforementioned theorem A is the C-algebra. By the definition this means, that there exists a locally compact zero-dimensional Hausdorff space X such that A is isomorphic with C∞ (X, F), where C∞ (X, F) is the subspace of all f ∈ Cb (X, F) for which for each ε > 0 there exists a compact subset Xε, f of X with | f (x)| < ε for each x ∈ X \ Xε, f . Recall that if A is commutative Banach algebra over K, then the spectrum of A denoted by Sp(A) is the set of all non-zero algebra homomorphisms from A into a field K topologized as a subset of K A . In accordance with Theorem 6.3 [Roo78] if X is a locally compact zero-dimensional Hausdorff space, then (Si) every a ∈ X induces an aˆ ∈ Sp C∞ (X) so that a( ˆ f ) := f (a), where f ∈ C∞ (X); (Sii) for every closed ring ideal I in C∞ (X) there exists a closed subset Z ⊂ X such that I = { f ∈ C∞ (X) : f = 0 on Z}. In particular, each closed ring ideal is an algebra ideal; (Siii) to every maximal ring (or algebra) ideal M in C∞ (X) there corresponds a unique a ∈ X such that M = { f ∈ C∞ : f (a) = 0}, each maximal ring (or algebra) ideal of C∞ (X) is the kernel of a unique homomorphism from C∞ (X) into K; (Siv) the mapping a 7→ aˆ is a homeomorphism of X onto Sp C∞ (X); (Sv) suppose that f ∈ C∞ (X), if X is compact, then Sp( f ) = f (X); otherwise Sp( f ) = f (X)∪{0}. In either case, Sp( f ) = f (X) and k f k = k f ksp , where kxksp := supφ∈Sp(A) |φ(x)| for an element x ∈ A in a commutative Banach algebra A. In view of this theorem we get that each maximal ideal B of C∞ (X, F) has the form B = { f ∈ C∞ (X, F) : f (z0 ) = 0}, where z0 is a marked point in X. On the other hand, as it was proved above F j 6= H j for each j ∈ No , hence there exists the following embedding c0 (F) ֒→ (H/F ) and this implies that (H/F )/c0 (F) is isomorphic with (H/c0 (F))/(F /c0 (F)). Therefore, H/F is isomorphic with c0 (F).

4.3.

Comments

Another methods of construction of isometrical representations of topological totally disconnected groups which may be non-locally compact with the help of quasi-invariant Fvalued measures were given in [Lud98b, Lud02b, Lud00a, Lud99t, Lud01s, Lud0348, Lud01f, LD03, Lud08] and references therein. One may mention that the non-local compactness of groups causes a twisted algebraic structure of measure spaces. This situation can be compared with representation theory of groups in non-Archimedean linear spaces. Over infinite fields with nontrivial non-Archimedean multiplicative norms the aforementioned theorem of Gelfand and Mazur is not accomplished due to existence of transcendental extensions of such fields (see Chapter 6 in [Roo78] and references therein). It was one of the basic reasons why there was demonstrated that even infinite compact groups and even commutative may have infinite-dimensional topologically irreducible representations in Banach spaces over non-Archimedean fields, for example, the additive group Zp of p-adic integer numbers [Dia79, Dia84, Dia95, R84, Roo78, RS71, RS73].

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The idea is the following (it was also communicated by B. Diarra). Take an algebra of bounded operators on a Banach space X over a locally compact infinite field K, consider its closed subalgebras A2 ⊂ A1 ⊂ L(X, X) such that A2 is a maximal closed ideal in A1 , but the quotient algebra A1 /A2 is isomorphic with a field which is a transcendental extension F of K. Let α be a transcendental element of F over K. Put T1 := β, where β ∈ θ−1 (α) is an invertible operator and θ : A1 → A1 /A2 is the quotient mapping. This induces a strongly continuous representation of the additive group Z so that Tn = T1n for each n ∈ Z. As usually GL(X) denotes the group of all invertible operators on X bounded together with its inverse, GL(X) ⊂ L(X, X). For suitable F and K and A1 and A2 , for example, when the residue class field of F contains the finite field Fp consisting of p elements, we can choose T : Z → GL(X) so that limn→∞ Tpn x = x for p-adic integers of the form pn , for each x in an infinite dimensional closed subspace Y of X on which A1 acts irreducibly. Then T has a strongly continuous extension on Zp . For this also the exponential and logarithmic functions for F can be used relating multiplicative and additive representations. Such representation may happen to be isometrical so that Tm ∈ IS(X), where IS(X) denotes the group of linear isometries of X. The considered representation has a topologically irreducible infinite dimensional component T : Z p → GL(X). That is clX spanK {Tg x : g ∈ Zp } = Y for each non-zero element x from some closed linear subspace Y in X, where Y is infinite dimensional over the field K. Mention also in relation with this that each ultra-metric space can isometrically be embedded into an infinite field with a non-trivial non-Archimedean multiplicative norm in accordance with Theorem 1.10 [Sch84]. The main feature of representations T : G → GL(H) of a group G in vector spaces H over a field K as it is well-known consists in using group algebras A(G) and the linear structure of H so that one has already not only a group homomorphism, but also the second operation related with addition of vectors Tg (ax + by) = aTg x + bTg y for all a, b ∈ K and x, y ∈ H and g ∈ G, where GL(H) ⊂ L(H, H) (see [Nai68, FD88]). For infinite locally compact Hausdorff topological groups the cornerstone for decomposition of unitary representations into direct integrals of topologically irreducible representations consists in using Haar measures and Banach algebras on groups associated with such measures. In the class of non-locally compact commutative groups there exist groups G having no any finite-dimensional topologically irreducible unitary representation T : G → U(H), but having infinite dimensional strongly continuous representations T : G → GL(H), where GL(H) is the general linear group on a Banach space H and U(H) is the unitary group on the complex Hilbert space H. That is invariant closed subspaces in H are all infinitedimensional in the considered case. This means that the closure of the F-linear span cl spanF {Tg x : g ∈ G} is infinite-dimensional over the field F for each non-zero vector x 6= 0 in the linear space H over a field F [B83, B87, B91]. Such groups can be constructed even as quotient groups of additive groups of topological vector spaces. In accordance with Theorem 5 [B87] if E is an infinite dimensional vector space over R with the topological weight of E equal to the topological weight of its topological dual space E ∗ , then there exists a discrete subgroup Z in E such that G := E/Z has no any non-trivial continuous character. This imply that such group has not any weakly continuous finite dimensional unitary representation, since it is Abelian and any weakly continuous finite dimensional unitary representations decomposes into the direct sum of

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characters corresponding to one-dimensional irreducible components. A topological group G is called bounded, if for each open neighborhood V of its unit element e there exists a natural number n ∈ N so that V n = G. Particularly, if G = E/Z, where Z is an additive subgroup of a normed space E, then G is bounded if there exists a positive number 0 < r < ∞ so that E = Z + rB, where B denotes the unit ball with center at zero in E. By Proposition 3 [B87] if G is a bounded commutative topological group satisfying the first axiom of countability and T is a weakly continuous representation of G in a complex Hilbert space, then T is equivalent to a unitary weakly continuous representation. Lemma 4 in [B83] states that every bounded representation of an amenable group in a Hilbert space is equivalent to a unitary representation, where a group G is called amenable if it possess an invariant left mean on the space Cb0 (G, C) of all continuous bounded functions on G supplied with the norm k f kC0 := supx∈G | f (x)|. b At the same time each topological group G admits a strongly continuous representation in a suitable Banach space S, for example, T : G → BUC(G, C) so that Tg 6= I for each g 6= e, by left shifts on the Banach space BUC(G, C) of bounded uniformly continuous complex-valued functions on G [B91]. If a Banach space H can be embedded into a Hilbert space X and this representation can weakly continuously be extended into X and if G is bounded, then this representation would be equivalent to unitary. Also when the group G is a dense subgroup in G1 and µ is a quasi-invariant non-negative non-trivial measure on G1 relative to left shifts from G, then G has a strongly continuous infinite dimensional unitary representation as it was outlined in [Lud06, Lud08]. This theory is rather complicated and is not considered in this book, but it is worth to note, that the non-commutative non-associative structure of measure and function algebras explain in part the known differences in the linear representation theory in nonArchimedean spaces and of non-locally compact groups in complex Hilbert spaces. Moreover, one gets more differences for representations of groups in linear spaces over infinite fields with non-Archimedean multiplicative norms. Indeed, in these cases any technique related with associative Banach algebras of functions on groups already does not work, since the algebras of functions on non-locally compact groups associated with quasi-invariant measures are already non-associative (see Lemmas 17 in Chapter III and 18 in Chapter IV) and the Gelfand-Mazur theorem is not valid for them. The initial algebras and quotient algebras as well in Theorems III.21 and IV.22 are non-associative and non-commutative and infinite-dimensional over the corresponding fields. This is logical also due to the following. In accordance with the A. Weil’s theorem if a Hausdorff group has a quasi-invariant non-trivial measure relative to itself, then it is locally compact (see [Bou63-69, FD88, VTC85] and Corollary 11 in Chapter III). Its analog for measures with values in non-Archimedean fields is valid as well (see Corollary 9 in Chapter IV). Recall that a commutative Hausdorff topological group is called exotic if does not admit any non-trivial strongly continuous unitary representation; and strongly exotic, if it does not admit any non-trivial weakly continuous representation (not necessarily unitary) in Hilbert spaces. Using technique of exotic groups it is possible to construct further examples of Banach-Lie commutative groups having infinite dimensional strongly continuous topologically irreducible unitary representations. For wrap (particularly loop) groups and groups of diffeomorphisms they were considered in [Lud06] besides cited above works.

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At first we mention the following well-known fact. C.1. Lemma. For an infinite dimensional complex Hilbert space X the entire unitary group U(X) is not topological group relative to the weak topology τw . Proof. It is sufficient to prove this lemma for the separable complex Hilbert space l2 (C), since l2 (C) has the embedding into X and inevitably U(l2 (C)) has an embedding into U(X). The weak operator topology τw is generated by base of neighborhoods of zero Wb (x1 , . . . , xn ; y1 , . . . , yn ) := {S ∈ L(X, X) : |(Sx j , y j )| < b ∀ j = 1, . . . , n} with 0 < b < ∞, n ∈ N, where L(X, X) denotes the space of all continuous linear operators from X into X; while (∗, ∗) denotes the scalar product in X, x1 , . . . , yn ∈ X. If (U(X), τw ) would be a topological group, then due to continuity of the multiplication (T, S) 7→ T S and inversion T 7→ T −1 and the conditions T ∗ T = T T ∗ = I for each T ∈ U(X) it would be closed in (B, τw ), where B := B(L(X, X), 0, 1) denotes the unit ball in L(X) := L(X, X) relative to the operator norm topology τn . Due to the Alaoglu-Bourbaki theorem (B, τw ) is compact [NB85]. Suppose that Un is a Cauchy net in U(X). In view of compactness of B there exists the limit limn Un =: A relative to the weak topology so that A ∈ B. If (U(X), τw ) is the topological group, then there exists limn Un∗Un = limn Un∗ limn Un = A∗ A = I, since Un∗Un = I for each n and the multiplication is continuous. Analogously limn UnUn∗ = limn Un limn Un∗ = AA∗ = I. Thus this would imply that (U(X), τw ) is complete and hence closed in B. Therefore, (U(X), τw ) would be also compact. As the compact topological group it would have a non-trivial nonnegative Haar measure λ on B f (U(X), τw ) [Bou63-69, FD88, Nai68]. Since X is separable, then B f (U(X), τw ) = B f (U(X), τn ). But U(X) is the Banach C∞ -manifold as well [Kl82], hence its tangent space TeU(X) is the separable Banach space, since (U(X), τn ) is separable, where τn is inherited from L(X). Each element in U(X) lies on some one-parameter subgroup [Bou76, Kl82, FD88, RS72]. To each local one-parameter subgroup some vector in Y := TeU(X) corresponds. Consider the exponential mapping from an open neighborhood V of zero in the algebra Y onto an open neighborhood U of the identity element e in U(X). Then the linear term of exp(tv) induces the shift w 7→ w + tv in Y , where v,tv ∈ V , w ∈ Y , t ∈ R. Consider embeddings U(Cn ) into U(X) for each n ∈ N and the multiplication of basic generators in TeU(Cn ) for each n. The multiplications in U(X) and in Y are related by the Campbell-Hausdorff formula locally on sufficiently small neighborhoods U and V [Bou76]. Consider the operator Lh g := hg in U(X), h, g ∈ U(X). It induces the operator Lh in Y . The multiplication in Y is [v, w] = ad v(w). But generally operators neither DLh − I nor (ad v) − I are compact in Y , v ∈ V , where I denotes the unit operator in Y . Nevertheless their compactness is the necessary condition for a quasi-invariance of a non trivial measure relative to Lh or ad v in Y (see Chapter I). Over R the space Y is isomorphic with l2 (R). Thus a measure on U(X) induces a measure ν on a neighborhood V = −V of zero in its tangent space Y so that 0 < ν(V ) < ∞. Since Y is separable then a measure on V induces the measure µ(A) := ∑∞j=1 ν(A ∩ (V + x j ))/2 j for each Borel subset A in Y , where {x j : j} is a set of vectors in Y so that x1 = 0 S and ∞j=1 (x j + V ) = Y . From the left invariance of λ relative to the entire U(X) it would follow that µ is quasi-invariant on Y relative to Y , but it can be quasi-invariant relative to neither Lh nor ad v, where h ∈ U, v ∈ V . But this is impossible due to Theorem I.3.18 and Corollary I.3.19. This contradiction finishes the proof.

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This can also be demonstrated directly using operators. For non-Archimedean Banach spaces an analogous result is valid. C.2. Lemma. For an infinite dimensional Banach space X over infinite locally compact field F with a non-Archimedean non-trivial multiplicative norm the entire isometry group IS(X) is not topological group relative to the weak topology τw . Proof. It is sufficient to prove this lemma for X of separable type over F, since if X ֒→ X1 is the embedding of Banach spaces, then there exists the embedding IS(X) ֒→ IS(X1 ) of groups. For the Banach space X the topologically dual space X ∗ separates points in X, since F is locally compact and so spherically complete [NB85, Roo78]. A base of neighborhoods of the weak topology τw in L(X, X) is: W (b; x1 , . . . , xn ; y1 , . . . , yn ) := {A ∈ L(X, X) : |y j (Sx j )| < b}, where 0 < b < ∞, x1 , . . . , xn ∈ X, y1 , . . . , yn ∈ X ∗ , n ∈ N. Each isometric operator S is characterized by the condition: kSxkX = kxkX for each x ∈ X. Therefore, if (IS(X), τw ) would be a topological group, then we shall show that it would be closed in (B, τw ), where B := B(L(X, X), 0, 1) is the unit ball in L(X, X) relative to the operator norm. By the Alaogly-Bourbaki theorem [NB85] (B, τw ) is compact for the locally compact field F. Consider a Cauchy net Un in (IS(X), τw ). Since (B, τw ) is compact, then there exists limn Un = A ∈ B relative to the weak topology. In view of the Hahn-Banach theorem (8.4.7) [NB85] for each x ∈ X there exists y ∈ X ∗ in the topological dual space X ∗ so that |y(x)| = |x|. Thus for each x ∈ X and each Un there exists y ∈ X ∗ so that |y(Un x)| = |x|, since |Un x| = |x| for each x ∈ X and Un ∈ IS(X). Let x ∈ X be non-zero, take 0 < ε < |x|. For any y ∈ X ∗ and each such ε and x there exists n0 such that |y(Un x) − y(Ax)| < ε for all n > n0 . On the other hand, supy∈X ∗ ,|y|=1 |y(Un x)| = |Un x| = |x|, since |y(z)| ≤ |y||z| for each z ∈ X and y ∈ X ∗ . Take particularly y such that |y(Ax)| = |Ax|, then |y(Un x) − y(Ax)| < ε for each n > n0 = n0 (x, y), hence |y(Un x)| = |Ax|, consequently, |Ax| = |x|, since A ∈ B ⊂ L(X, X) and y ◦ A ∈ X ∗ and Ax ∈ X and also limn [limm y(Un−1Um z)] = limn y(Un−1 Az) = limn y(AUn−1 z) = y(z) for all y ∈ X ∗ and z ∈ X. Thus (IS(X), τw ) would be complete and hence closed in B. Therefore, as the compact group (IS(X), τw ) would have a Haar R-valued on B f (IS(X), τw ) = B f (IS(X), τn ) and also Ks -valued measures on Bco(IS(X)), τw ), where s 6= p, while the residue class field of F contains the finite field Fp . Then consider the exponential mapping from an open neighborhood V of zero in the algebra Y onto an open neighborhood U of the identity element e in IS(X). We get that the linear term of exp(tv) induces the shift w 7→ w + tv in Y , where v,tv ∈ V , w ∈ Y , t ∈ F. There exist embeddings IS(F n ) into IS(X) for each n ∈ N and the multiplication of basic generators in Te IS(F n ) for each n. The multiplications in IS(X) and in Y are related by the Campbell-Hausdorff formula locally on sufficiently small neighborhoods U and V [Bou76]. Consider the left multiplication operator Lh g := hg in IS(X), h, g ∈ U(X). It induces the operator Lh in Y . The multiplication in Y is [v, w] = ad v(w). But generally operators neither DLh − I nor (ad v) − I are compact in Y , v ∈ V , where I denotes the unit operator in Y . Nevertheless their compactness is the necessary condition for a quasi-invariance of a non trivial measure relative to Lh or ad v in Y (see Chapters I and II). The isometry group IS(X) is the Banach manifold as well and its tangent space Te IS(X) is isomorphic with the Banach space of separable type over F [Bou76, Roo78]. Using Theorem II.3.13 and Corollary II.3.14 for the Ks -valued measure or Theorem I.3.18 and

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Corollary I.3.19 for the R-valued measure analogously to § C.1 we infer the statement of this lemma. Nevertheless, for a complex Hilbert space X weak and strong continuity of a unitary representation T : G → U(X) of a topological group G are equivalent, but this is another thing, because it is already supposed that G is the topological group [Nai68, FD88]. Recall that a locally convex representation T of a topological group G in a locally convex space X over a field F is called topologically irreducible, if T : G → GL(X), Tg 6= I for some g ∈ G, and there is not any closed F-linear subspace other than {0} and X stable (invariant) under T , that is Tg X ⊂ X for each g ∈ G. C.3. Theorem. For each infinite dimensional real Hilbert space E2 there exists a discrete additive subgroup H2 in E2 such that E2 /H2 has no any continuous character, but has infinite dimensional topologically irreducible strongly continuous unitary representations. The family of such pairwise non-equivalent representations is at least c := card(R). Proof. It is sufficient to demonstrate this theorem for the separable Hilbert space E. If X is a non-separable infinite dimensional Hilbert space, then X = l2 ⊕Y , where Y = X ⊖ l2 and there is the natural projection operator P : X → l2 , P(x + y) = x for each x ∈ l2 and y ∈ Y . If X and l2 are considered as additive groups this gives the homomorphism. In accordance with [B91] there exists a discrete subgroup H in X so that X/H is exotic and H = H1 ⊕ H2 , where H1 = H ∩ l2 and H2 = Y ∩ H up to an isomorphism of Hilbert spaces. Then P(H) = H1 and P induces the quotient mapping from X/H onto l2 /H1 , so that l2 /H1 is also exotic. Remind that a symmetric Hilbert-Schmidt non-degenerate operator A in the separable real Hilbert space E2 has an orthonormal system of vectors { f j : j ∈ N} and numbers s j ∈ R such that A f = ∑ j s j ( f , f j ) f j for each f ∈ E2 , where ∑ j s2j < ∞. For the nuclear (trace) operator ∑ j |s j | < ∞. If A is positive definite, then s j > 0 for each j. If λ j are eigenvalues of such positive symmetric nuclear operator A = A∗ > 0, then 0 < ∑ j λ j ≤ ∑ j s j . The product of two Hilbert-Schmidt operators is the nuclear operator [Pie65]. Let now E be separable and H be its discrete subgroup so that E/H is the exotic group (see Theorems 5.3 and 6.1 [B91]). The procedure of a choice of H described there is inductive related with a consideration of volumes of some convex bodies in linear subspaces in E, so that each generator v j of H on j-th step is chosen from some suitable open subset in the Euclidean subspace Rn(j) embedded into E. Construct the rigged Hilbert space E2 ֒→ E1 ֒→ E with linear embeddings θ1 : E1 → −1 E and θ2 : E2 → E1 such that A−1 1 : E → E1 and A2 : E1 → E2 , where A1 and A2 are symmetric non-degenerate positive definite operators of Hilbert-Schmidt class, hence A = A1 A2 is the trace class (nuclear) operator, choose these operators satisfying A2 = A1 |E2 with kA2 kL(E2 ,E1 ) ≤ 1 and kA1 kL(E1 ,E) ≤ 1 [DF91]. The scalar products in these spaces −1 −1 −1 are such that (x, y)E1 = (A−1 1 x, A1 y)E for each x, y ∈ E1 and (x, y)E2 = (A2 x, A2 y)E1 = (A−1 x, A−1 y)E for all x, y ∈ E2 , where (∗, ∗)E2 , (∗, ∗)E1 and (∗, ∗)E denote scalar products in these Hilbert spaces E2 , E1 , E (see § § II.2.1 and 2.2 in [DF91]). Therefore, kθ(x)kE ≤ kθ2 (x)kE1 ≤ kxkE2 for each x ∈ E2 , where θ(x) = θ1 (θ2 (x)) for all x ∈ E2 . Choose H contained in E2 , that is possible since H is discrete relative to k ∗ kE , hence H is discrete in E1 and E2 relative to their norms. Therefore, θ j induce embeddings θ1 : E1 /H1 → E/H and θ2 : E2 /H2 → E1 /H1 and their composition θ : E2 /H2 → E/H, where H1 := θ2 (H2 ), H = θ1 (H1 ) = θ(H2 ).

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The cylinder Gaussian distribution on E1 with the unit correlation operator and zero mean value (centered) induces the Gaussian σ-additive probability measure ν on E quasiinvariant relative to E1 . The correlation operator of such Gaussian measure is A1 . This induces the measure µ on E/H quasi-invariant relative to E1 /H1 for suitable correlation operator A1 . If Q is a νy -null subset in E with a marked y ∈ E1 , then νz (Q + H) = 0 for each z ∈ E1 , since ν is countably additive and quasi-invariant relative to shifts on vectors from E1 . For each Borel subset Q in E/H its counter-image q−1 (Q) is a Borel subset in E so that ν(q−1 (Q)) = µ(Q), where q : E → E/H is the quotient mapping. Thus, µa ≪ µb for all a, b ∈ E1 /H1 and hence µa ∼ µ for each a ∈ E1 /H1 , since E1 /H1 is the additive dense subgroup embedded into E/H. Consider an arbitrary open subset V in E such that V ∩ H ⊂ {0}, which is possible, since H is discrete in E. Then in E/H its image is V + H. Therefore, ν(V + H) =

∑ ν(V + h) and

(1)

h∈H

µa (dx)/µ(dx) = [µa (dx)/ν(dy)]/[µ(dx)/ν(dy)]      = ∑ νb+h (dy)/ν(dy) ∑ νh (dy)/ν(dy) h∈H

=

h∈H



∑

h∈H

    ρν (h + b, y) ∑ ρν (h, y) ,

(2)

h∈H

where a = b + H, b ∈ E2 , x = y + H, b and y are vectors in E2 of minimal absolute values k ∗ kE2 satisfying these conditions. In accordance with Theorem I.4.2 [DF91] −1/2

ρν (h + b, y) = exp{(A−1 1 (h + b), y) − (A1

−1/2

(h + b), A1

(h + b))/2},

where (∗, ∗) = (∗, ∗)E is the scalar product in E, hence series in (2) are converging and continuous on G × (E/H), where G := E2 /H2 (see Remark I.4.1 [DF91]). Indeed, the function ω(b, y) :=

∑ exp{(A−1 1 (h + b), y) − (A1

−1/2

−1/2

(h + b), A1

(h + b))/2}

h∈H

is positive and continuous on E2 × E due to the Weierstrass theorem and estimates of the remainder due to the generalization of Cauchy integral theorem for series of positive addends and using σ-additive Gaussian measures on E2 up to positive multipliers. Moreover, the latter function ω(b, y) on product of bounded balls in E2 × E is bounded from below, since infh∈H\0 khk > 0. Consider independent generators {v j : j ∈ N} of H2 so that each h ∈ H2 is the linear combination of v j with integer expansion coefficients m j ∈ Z, where v j 6= 0 for each j. Suppose that for A2 eigenvectors w j are given with eigenvalues σ2j , A2 w j = σ2j w j . Certainly clE2 spanC {w j : j} = E2 . Then  n  2 2 2 ρν (h + b, y) = lim exp ∑ [(yk (hk + bk )/σk ) − (hk + bk ) /(2σk )] , (3) n→∞

k=1

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where yk denotes coordinates of y in the basis vk /kvk kE , h ∈ H2 , b ∈ E2 . Consider strongly continuous unitary regular representation of G := E2 /H2 in the Hilbert space L2 (E/H, µ, C) =: X associated with the quasi-invariant measure µ (see [Lud02b, Lud06, Lud08]). Remind its construction. The scalar product in X is R ¯ ( f , g) := E/H f (y)g(y)µ(dy), where z¯ denotes the complex conjugated number z, f : E/H → C. Take a marked number b ∈ R, i = (−1)1/2 , and put ib+1/2

Th f (y) := ρµ

(h, y) f (h−1 y)

for each h ∈ G and f ∈ X, where ρµ (h, y) = µh (dy)/µ(dy), µh (A) := µ(h−1 A) for each Borel subset A in E/H and every h ∈ G. Suppose that T : G → U(X) has a finite dimensional reducible component. Each finite dimensional unitary representation decomposes into direct sum of characters up to an intertwining operator on X. We demonstrate that each continuous character ψ : G → S1 , where S1 is the unit circle in C, gives the continuous character χ := ψ ◦ θ−1 on E/H. Indeed, to each non-trivial continuous characters ψ and also χ there correspond continuous linear functionals f2 and f on E2 and E respectively so that f |E2 = f2 is continuous, since the topology of E2 is stronger than that of E (see [Eng86] and Proposition 4.5 in [B91]). For each linear continuous functional f2 on a Hilbert space E2 there exists a decomposition E2 = L1 ⊕ M2 , where L1 is a one-dimensional R-linear subspace in E2 and M2 = ker( f2 ) = f2−1 (0) (see § III.1.6 [KF89]). Since E2 is the subspace in E, then L1 ⊂ E. Take the orthogonal complement M of θ(L1 ) in E so that E = L1 ⊕ M. Put f (x) := f2 (x1 ), where x = x1 + m is the unique decomposition of each x ∈ E with x1 ∈ L1 and m ∈ M, since E is presented as the direct sum of these subspaces L1 and M. Therefore, f is the declared continuous extension of f2 from E2 onto E and inevitably there exists the continuous character χ extending the character ψ from G onto E/H. The space E2 is everywhere dense in E, consequently, E2 is not contained in M. Therefore, if f2 is non-trivial, then f is non-trivial. But E/H is exotic, hence f is trivial and inevitably f2 is trivial, that produces the contradiction. Thus G has no any continuous character and no any weakly continuous finite dimensional unitary representation. It remains that it has the infinite dimensional strongly continuous topologically irreducible unitary representation. By our construction E2 is the separable Hilbert space, hence isomorphic with the standard separable Hilbert space, while H2 is its discrete subgroup. If E3 is not a separable Hilbert space, then there exists the quotient mapping P : E3 /H3 → E2 /H2 induced by the projection P from E3 onto E2 with P(H3 ) = H2 , where E2 is separable and closed and embedded into E3 , while H3 is a discrete additive subgroup in E3 , H2 = H3 ∩ E2 , E3 = E2 ⊕ E4 , where E4 = E3 ⊖ E2 , H3 = H2 + H4 , H4 = H3 ∩ E4 , E2 ⊥ E4 . Thus any infinite dimensional topologically irreducible unitary representation T of G induces the claimed representation T ◦ P for E3 /H3 . Two unitary representations associated with quasi-invariant measures are equivalent if and only if the corresponding measures are equivalent (see [Lud06, Lud08]). The family of pairwise inequivalent quasi-invariant Gaussian measures on E and hence on E/H is at least c due to Theorems about equivalence and orthogonality of Gaussian measures [DF91, Sko74]. Thus G is the desired additive group.

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Mention that the topology τE/H of E/H induces the topology τ′ in G strictly weaker, than the initial topology τG of G and the quasi-invariance factor ρµ (a, x) does not exist for all a ∈ E/H, consequently, the regular representation T = T µ of (G, τG ) has not any weakly continuous extension on (E/H, τE/H ) (see also Lemmas III.12 and IV.13). C.4. Corollary. For each infinite dimensional real Hilbert space there exists a family of topologically irreducible infinite dimensional strongly continuous unitary representations. This family has at least c pairwise inequivalent representations. Proof. Take E2 and the quotient mapping q : E2 → E2 /H2 = G from §C.3. Each strongly continuous infinite dimensional unitary representation T : E2 /H2 → U(X) induces the unitary representation T ◦ q =: S : E2 → U(X). As the composition of continuous mappings it is continuous. Moreover, S(H2 ) = I, hence if T is topologically irreducible, so S also is such. Suppose that G is an infinite topological group with a topology τ such that a topological density of G is d(G, τ) ≥ ℵ0 and its topological character is χ(G, τ) ≥ ℵ0 (see also [Eng86]). Let also µ be a quasi-invariant non-trivial regular Radon measure on G with values in either F = R or in the non-Archimedean field F = Ks . Consider an everywhere dense subset J in G of the cardinality card(J) = d(G, τ). For each point x ∈ J its base U x of open neighborhoods can be chosen of cardinality χ(x, τ) = χ(G, τ), since G is the topological group. Then the density of either Xτ = Lr (G, µ, C) with 1 ≤ r < ∞ or Xτ = L(G, µ, Ks ) correspondingly is d(Xτ , k ∗ kXτ ) = d(G, τ)χ(G, τ)d(F, | ∗ |F ), since finite linear combinations of characteristic functions of open subsets U j from the family {U x : x ∈ J} are dense in this Banach spaces due to Radon property of µ and its regularity and that each infinite covering of a compact subset in G by U j has a finite sub-covering. Particularly if G is metrizable and F is locally compact, then d(Xτ , k ∗ kXτ ) = d(G, τ)ℵ20 = d(G, τ). Thus if the topological group G is supplied with two topologies τ and ξ and measures µ and ν as above are so that ℵ0 ≤ d(G, τ)χ(G, τ) < d(G, ξ)χ(G, ξ) and d(F, | ∗ |F ) ≤ d(G, τ)χ(G, τ), then d(Xτ , k ∗ kXτ ) < d(Xξ , k ∗ kXξ ). This implies that Xξ and Xτ are not isomorphic Banach spaces in the aforementioned cases. Therefore, a problem of finding topologically irreducible components of strongly continuous representations of (G, τ) in Xτ can generally not be reduced to that of (G, ξ). For example, when (G, τ) is a metrizable separable Frechet-Lie non-discrete topological group over K, then card(G) ≥ card(K) ≥ c := card(R) > ℵ0 := card(N), but d(G, τ) = χ(G, τ) = ℵ0 . When d(F, | ∗ |F ) is greater, than ℵ0 it is possible to take a locally compact subfield K in F and consider a K-linear Banach space XK so that the enlargement of the field from K to F gives from XK the Banach space X over F that to judge whether two spaces Xτ and Xξ are linearly topologically isomorphic or not. If now ξ is the discrete topology in G, then d(G, ξ) = card(G) > d(G, τ)χ(G, τ) = ℵ20 = ℵ0 . Moreover, topologies τ and ξ in G may generally be incomparable and a representation strongly continuous relative to one topology may be discontinuous relative to another topology. Mention that an algebraic reduction of a representation implies that a group is considered relative to the discrete topology. Remind that a topological space Q is called locally compact, if for each q ∈ Q there exists a neighborhood U of q such that cl(U) is the compact subspace in Q. For compact and locally compact groups C∗ -algebras are used, where

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by the definition C∗ -algebras are complete. On C∗ -algebras multiplicative continuous linear functionals are taken and related closed ideals are considered [FD88]. For either compact or a locally compact group G and a weakly continuous unitary representation T : G → U(X) some tricks are used. They are caused by the facts: (i) that a continuous image of a compact set is compact, (ii) each continuous bijective mapping of the compact space on a Hausdorff space is a homeomorphism, (iii) each uniformity on the compact topological space is complete (see Theorems 3.1.10, 3.1.13, 8.3.15 in [Eng86] and [FD88, HR79]). This also is based on completeness of C∗ -algebras and their ideals. Indeed, a topological group is T0 if and only if it is completely regular (see Theorems 4.8 and 8.4 in [HR79]), but a topological space is T1 if and only if each its point is closed in it (see § 1.5 in [Eng86]). A subset Q of a complete uniform space Y is complete if and only if Q is closed in Y in accordance with Theorem 8.3.6 [Eng86]. Therefore, if f : A → B is a quotient mapping of algebras, where B is supplied with the quotient topology, then B is T0 or Hausdorff if and only if f −1 (0) is closed in A. So that a quotient of A by a non-closed ideal produces a non-Hausdorff even without any separability axiom algebra. But this technique generally is useless for non locally compact groups. Indeed, the unitary group U(X) of an infinite dimensional complex Hilbert space is not closed in the algebra L(X, X) relative to the weak topology τw in L(X, X) so that the weak closure of U(X) is not a group [Nai68]. For a non locally compact topological group its image T (G) in (L(X, X), τw ) need not be complete and its completion need not be a group, moreover, (T (G), τw ) generally may be not a topological group even when T is bijective. Using results of this book it is possible to make further investigations of non-locally compact totally disconnected topological groups, their structures and representations, measurable operators in Banach spaces over non-Archimedean fields, apply this for the development of non-Archimedean quantum mechanics and quantum field theory, quantum gravity, superstring theory and gauge theory, etc. Certainly this measure theory is helpful for studying random functions and stochastic processes in Banach spaces and topological groups.

Appendix A

Operators in Banach Spaces Suppose X = c0 (ω0 , K) is a Banach space over a locally compact non-Archimedean infinite field K with a non-trivial normalization and I is a unit operator on X. If A is an operator on X, then in some basis of X we have an infinite matrix (Ai, j )i, j∈N , so we can consider its transposed matrix At . If in some basis the following equality is satisfied At = A, then A is called symmetric. A.1. Lemma. Let A : X → X be a linear invertible operator with a compact operator (A − I). Then there exist an orthonormal basis (e j : j ∈ N) in X, invertible linear operators C, E, D : X → X with compact operators (C − I), (E − I), (D − I) such that A = SCDE, D is diagonal, C is lower triangular and E is upper triangular, S is an operator transposing a finite number of vectors from an orthonormal basis in X. Moreover, there exists n ∈ N and invertible linear operators A′ , A” : X → X with compact operators (A′ − I), (A” − I) and (A′i, j − δi, j = 0) for i or j > n, A” is an isometry and there exist their determinants det(A′ )det(A”) = det(A), |det(A”)|K = 1, det(D) = det(A). If in addition A is symmetric, then Ct = E and S = I. Proof. In view of Lemma 2.2[Sch89] for each c > 0 there exists the following decomposition X = Y ⊕ Z into K-linear spaces such that k(A − I)|Z k < c, where dimKY = m < ℵ0 . In the orthonormal basis (e j : j) for which spanK (e1 , . . . , em ) = Y for c ≤ 1/p we get A = A′ A” with (A − I)|Z = 0, |A”i, j − δi, j | ≤ c for each i, j such that (A′i, j − δi, j ) = 0 for i or j > n, where n ≥ m is chosen such that |Ai, j − δi, j | ≤ c2 for i > n and j = 1, . . . , m, Ai, j := e∗i (Ae j ), e∗i are vectors ei considered as linear continuous functionals e∗i ∈ X ∗ . Indeed, (Ai, j : i ∈ N) = Ae j ∈ X and limi→∞ Ai, j = 0 for each j. From the form of A” it follows that kA”e j − e j k ≤ 1/p for each j, consequently, kA”xk = kxk for each x ∈ X. Since A” = (A′ )−1 A, (A − I) and (A′ − I) being compact, hence (A” − I) is compact together with (A−1 − I), ((A′ )−1 − I) and ((A”)−1 − I). Moreover, there exists limk→∞ det(A)k = det(A) = limk det((A′ )k (A”)k ) = limk det(A′ )k det(A”)k = det(A′ )det(A”), where (A)k := (Ai, j : i, j ≤ k). This follows from the decompositions X = Yk ⊕ Zk for c = c(k) → 0 whilst k → ∞. This means that for each c(k) = p−k there exists n(k) such that |Ai, j − δi, j | < c(k), |A′i, j − δi, j | < c(k) and |A”i, j − δi, j | < c(k) for each i or j > n(k), consequently,
1···n(k)i1 ···iq
1···n(k)
r |A 1···n(k) −A 1···n(k) δi1 , j1 · · · δiq , jq | < c(k), where A ij11 ···i ··· jr is a minor corresponding to j1 ··· jq rows i1 , . . . , ir and columns j1 , . . . , jr for r, q ∈ N. From the ultra-metric inequality it follows that |det(A”)−1| ≤ 1/p, hence |det(A”)|K = 1, det(A”)k 6= 0 for each k, det(A′ )k = det(A′ )n

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for each k ≥ n. Using the decomposition of det(A′
)n by the last row (analogously by the column) we get A′n, j 6= 0 and a minor A′ 1···1···n−1 j−1, j+1,n 6= 0. Permuting the columns j and n (or
′ 6= 0. Therefore, by the enumeration rows) we get as a result a matrix (A¯ )n with A¯ ′ 1···n−1 1···n−1
1···k ′ of the basic vectors we get A 1···k 6= 0 for each k = 1, . . . , n, since |det(A¯ ′ )n | = |det(A′ )n |.
j Therefore, there exists the orthonormal basis (e j : j) such that A 1··· 1··· j 6= 0 for each j
j and lim j A 1··· 1··· j = det(A) 6= 0. Applying to (A) j the Gaussian decomposition and using compactness of A − I due to formula (44) in § II.4[Gan88], in the of

which is valid
case
j 1··· j−1 1,...,k−1,g 1···k K also, we get D = diag(D j : j ∈ N), D j = A 1··· /A ; C = A /A g,k 1··· j 1··· j−1 1,...,k−1,k 1···k ;

1,...,k−1,k Ek,g = A 1,...,k−1,g /A 1···k 1···k for g = k + 1, k + 2, . . . , k ∈ N. Therefore, (C − I), (D − I), (E − I) are the compact operators, Ci, j , D j , Ei, j ∈ K for each i, j. Particularly, for At = A (At denotes the transposed matrix for A) we get Ek,g = Cg,k . A.2. Notes. 1. An isometry operator S does not influence on the results given above by the measures, since A|Z and S−1 A on X have decompositions of the form CDE, where X ⊖ Z is a finite-dimensional subspace of X over K, hence X ⊖ Z is locally compact for the locally compact field K. Therefore, on X ⊖ Z there exists the Haar measure with values in R or Qs with s 6= p, K ⊃ Qp . A.2.2. For compact A − I we can construct the following decomposition A = BDBt C, where BBt = I, CCt = I, D is diagonal, B, D and E are operators on c0 (ω0 , Cp ) with limi j →∞ (Bi, j − δi, j ) = lim j (D j − 1) limi+ j→∞ (Ci, j − δi, j ) = 0. But, in general, matrix elements of B,C, D are in Cp and may be in Cp \K, since from the secular equation det(A−λI) even for symmetric matrix A over K in general may appear p1/n for n ∈ N, but R is not contained in K and Cp is not locally compact. This decomposition is not used in the present chapter for the construction of quasi-invariant measures on a Banach space X over K, since on Cp there is not any non-trivial invariant measure (or even quasi-invariant relative to all shifts from Cp ) and may be Bi, j , D j , Ci, j ∈ Cp \ K. Instead of it (and apart from [Sko74]) we use the decomposition given in the Lemma A.1. A.3. Remarks. Let K be a non-Archimedean infinite field with a non-trivial normalization. Let X and Y be normed spaces over K, and F : U → Y be a function, where U ⊂ X is an open subset. The function F is called differentiable, if for each t ∈ K, x ∈ U and h ∈ X such that x + th ∈ U there exists DF(x, h) := {dF(x + th)/dt | t = 0} = limt→0,t6=0 {F(x +th) − F(x)}/t and DF(x, h) linear by h, that is, DF(x, h) =: F ′ (x)h, where F ′ (x) is a bounded linear operator (derivative). Let Φ1 F(x; h;t) := {F(x + th) − F(x)}/t for each t 6= 0, x ∈ U, x + th ∈ U, h ∈ X. If this function Φ1 F(x; h;t) has a continu¯ 1 F on U × V × S, where U and V are open neighborhoods of x and 0 ous extension Φ ¯ 1 F(x; h;t)k := sup{kΦ ¯ 1 F(x; h;t)k/khk : x ∈ U, 0 6= h ∈ V,t ∈ in X, S = B(K, 0, 1) and kΦ 1 ′ ¯ S} < ∞ and Φ F(x; h; 0) = F (x)h, then F is called continuously differentiable on U, the space of all such F is denoted by C1 (U,Y ), where B(X, y, r) := {z ∈ X : kz − ykX ≤ r}. By induction we define Φn+1 F(x; h(1), . . . , h(n + 1);t(1), . . . ,t(n + 1)} := {Φn F(x + t(n + 1)h(n + 1); h(1), .., h(n);t(1), . . .,t(n)) − Φn F(x; h(1), . . . , h(n);t(1), . . . ,t(n))}/t(n + 1) and the space Cn (U,Y ) [Lud99t, Lud98b]. The family of all bijective surjective mappings of U onto itself of class Cn is called the diffeomorphism group and it is denoted by Di f f n (U) := Cn (U,U) ∩ Hom(U). A.4. Theorem. Let K be a spherically complete non-Archimedean infinite field with non-trivial normalization, F ∈ Cn (U,Y ) with n ∈ N := {1, 2, ..}, F : X → Y , X and Y be Ba-

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nach spaces, U be an open neighborhood of y ∈ X, F(X) = Y and [F ′ (y)]−1 be continuous, where n ∈ Z, 0 ≤ n < p when char(K) =: p > 0, n ≥ 0 when char(K) = 0. Then there exists r > 0 and a locally inverse operator G = F −1 : V := B(Y, F(y), kF ′ (y)kr) → B(X, y, r), moreover, G ∈ Cn (V, X) and G′ (F(y)) = [F ′ (y)]−1 . Proof. Let us consider an open subset U ⊂ X, [y, x] := [z : z = y + t(x − y), |t| ≤ 1, t ∈ K] ⊂ U and g : Y → K be a continuous K-linear functional, f (t) := g(F(y + t(x − y)). Such family of non-trivial g separating points of X exists due to the Hahn-Banch theorem [NB85, Roo78], since K is spherically complete. Then there exists f ′ (t) = g(F ′ (y + t(x − y))(x − y)), f ∈ Cn (B(K, 0, 1), K), consequently, for F ′ (z) 6= 0, z ∈ [x, y] there exists b > 0 such that for g(F ′ (z)) 6= 0 we have g(F(x + t(x − y))) − g(F(x))| = |g(F ′ (z)(x − y))| × |t| for each |t| < b. From |g(u(z)) − g(u(x))| = |g(z) − g(x)| and the Hahn-Banach it follows, that u is the local isometry, where u(z) := [F ′ (y)]−1 (F(z)), z, x ∈ B(X, y, r), r is chosen ¯ 1 u(x, h,t)) − hk < khk/2 for each x ∈ B(X, y, r), h ∈ B(X, 0, r), t ∈ B(K, 0, 1). such that k(Φ Therefore, F(B(X, y, r)) ⊂ B(Y, F(y), kF ′ (y)kr) =: S. Applying to the function H(z) := z − [F ′ (y)]−1 (F(z) − q) for q ∈ S the fixed point theorem we get z′ such that F ′ (z′ ) = q, since kH(x) − H(z)k < b × kx − zk. From the uniform continuous by (g, h) differentiability fg,h (t) := g(F(y + th) by t it follows continuous differentiability of G. A.5. Theorem. Let f ∈ Cn (U,Y ), where n ∈ N, Y is a Banach space, n ∈ Z, 0 ≤ n < p when p := char(K) > 0, n ≥ 0 when char(K) = 0. Then for each x and y ∈ U is ac( j) j complished the following formula: f (x) = f (y) + ∑n−1 j=1 f (y)(x − y) / j! + Rn (x, y)(x − ¯ j f ) (y; x − y, . . . , x − y; 0, . . . , 0) × j!, f ( j) (y)h j := f ( j) (y) y)n−1 , where f ( j) (y)(x − y) j = (Φ (h, . . . , h), f ( j) (y) : U → L j (X ⊗ j ,Y ), Rn (x, y) : U 2 → Ln−1 (X ⊗(n−1) ,Y ) with Rn (x, y) = o(kx − yk), where L j (X ⊗ j ,Y ) is the Banach space of continuous polylinear operators from X ⊗ j to Y , U is open in X. ¯ 1 f . For n = 2 let us take Proof. For n = 1 this formula follows from the definition of Φ 2 n−1 n ¯ f (x; y, y). Evidently, C (U,Y ) ⊃ C (U,Y ). Let the statement be true for n − R2 (x, y) = Φ ¯ n−1 f (y +tn (x − y); x − y, . . . , x − y;t1 , . . . ,tn−1 ) = Φ ¯ n−1 f (y; x − 1, where n ≥ 3. Then from Φ 1 n−1 ¯ ¯ y, . . . , x − y;t1 , . . . ,tn−1 )+ (Φ (Φ f (y; x − y, . . . , x − y;t1 , . . . ,tn−1 ))(y; x − y;tn )) × tn and ¯ n it follows that Rn (x, y)(x − y)n−1 = Φ ¯ n f (y; x − y, . . . , x − y; 0, . . . , 0, 1)− continuity of Φ f (n) (y)(x − y)n /n! = α(x − y)(x − y)n−1 , where limx→y, x6=y α(x − y)/kx − yk = 0, that is, α(x − y) = o(kx − yk). A.6. Suppose that X and Y are Banach spaces over a (complete relative to its uniformity) locally compact field K. Let X and Y be isomorphic with the Banach spaces c0 (α, K) and c0 (β, K) and there are given the standard orthonormal bases {e j : j ∈ α} in X and {q j : j ∈ β} in Y respectively, then each E ∈ L(X,Y ) has its matrix realization E j,k := q∗k Ee j , where α and β are ordinals, q∗k ∈ Y ∗ is a continuous K-linear functional q∗k : Y → K corresponding to qk under the natural embedding Y ֒→ Y ∗ associated with the chosen basis, Y ∗ is a topologically conjugated or dual space of K-linear functionals on Y . A.7. Let A be a commutative Banach algebra and A+ denotes the Gelfand space of A, that is, A+ = Sp(A), where Sp(A) in another words spectrum of A was defined in Chapter 6 [Roo78]. Let C∞ (A+ , K) be the same space as in [Roo78, LD02]. A.8. Definition. A commutative Banach algebra A is called a C-algebra if it is isomorphic with C∞ (X, K) for a locally compact Hausdorff totally disconnected topological space X, where f + g and f g are defined point-wise for each f , g ∈ C∞ (X, K). A.9. Remark. Fix a Banach space H over a non-Archimedean complete field F, as

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above L(H) denotes the Banach algebra of all bounded F-linear operators on H. If b ∈ L(H) we write shortly Sp(b) instead of SpL(H) (b) := cl(Sp(spanF {bn : n = 1, 2, 3, . . . })) (see also [Roo78]). It was proved in Theorem 2 in [Put68] in the case of F with the discrete normalization group, that each continuous F-linear operator A : E → H with kAk ≤ 1 from one Banach space E into another H has the form ∞

A =U

∑ πn Pn,A ,

n=0

where Pn := Pn,A , {Pn : n ≥ 0} is a family of projections and Pn Pm = 0 for each n 6= m, kPn k ≤ 1 and Pn2 = Pn for each n, U is a partially isometric operator, that is, U|cl(∑n Pn (E)) is isometric, U|E⊖cl(∑n Pn (E)) = 0, ker(U) ⊃ ker(A), Im(U) = cl(Im(A)), π ∈ F, |π| < 1 and π is the generator of the normalization group of F. We restrict our attention to the case of the locally compact field F, consequently, F has the discrete valuation group. If kAk > 1 we get (i) A = λAU

∞

∑ πn Pn,A ,

n=0

where λA ∈ F and |λA | = kAk. In view of [LD02] this is the particular case of the spectral integration on the discrete topological space X. Evidently, for each 1 ≤ r < ∞ there exists J ∈ L(H) for which 1/r  r (ii) ∑ sn dimF Pn,J (H) < ∞ n≥0

for 1 ≤ r < ∞, where J has the spectral decomposition given by Formula (i), sn := |λJ ||π|n kPn k. Using this result it is possible to give the following definition. A.10. Definition. Let E and H be two normed F-linear spaces, where F is an infinite spherically complete field with a nontrivial non-Archimedean normalization. The F-linear operator A ∈ L(E, H) is called of class Lq (E, H) if there exists an ∈ E ∗ and yn ∈ H for each n ∈ N such that   ∞ q q (i) ∑ kan kE ∗ kyn kH < ∞ n=1

and A has the form

∞

(ii) Ax =

∑ an (x)yn

n=1

for each x ∈ E, where 0 < q < ∞. For each such A we put  ∞ 1/q

q q

, (iii) νq (A) = inf ∑ an kE ∗ kyn kH n=1

where the infimum is taken by all such representations (ii) of A, (iv) ν∞ (A) := kAk and L∞ (E, H) := L(E, H).

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A.11. Proposition. Lq (E, H) is the normed F-linear space with the norm νq , when 1 ≤ q; it is the metric space, when 0 < q < 1. Proof. Let A ∈ Lq (E, H) and 1 ≤ q < ∞, since the case q = ∞ follows from its definition. Then A has the representation A.10.(ii). Then due to the ultra-metric inequality kAxkH ≤ kxkE sup(kan kE ∗ kyn kH ) ≤ kxkE n∈N



∞

∑

q q kan kE ∗ kyn kH

n=1

1/q

,

hence supx6=0 kAxkH /kxkE =: kAk ≤ νq (A). Let now A, S ∈ Lq (E, H), then there exists 0 < δ < ∞ and two representations Ax = ∞ ∑∞ n=1 an (x)yn and Sx = ∑m=1 bm (x)zm for which  

∞

∑

q q kan kE ∗ kyn kH

n=1

∞

∑

q q kbn kE ∗ kzn kH

n=1

(A + S)x = ∑∞ n=1 (an (x)yn + bn (x)zn ) and νq (A + S) ≤



∞

∑ kan k kyn k

n=1

q

q

1/q

+

1/q

1/q



∞

≤ νq (A) + δ and

≤ νq (S) + δ, hence

∑ kbn k kzn k

n=1

q

q

1/q

≤ νq (A) + νq (S) + 2δ

due to the H¨older inequality. The case 0 < q < 1 is analogous to the classical one given in [Pie65]. A.12. Proposition. If J ∈ Lq (H), S ∈ Lr (H) are commuting operators, the field F is with the discrete valuation group and 1/q + 1/r = 1/v, then JS ∈ Lv (H), where 1 ≤ q, r, v ≤ ∞. Proof. Since F is with the discrete valuation, then J and S have the decompositions A.9.(i). Certainly each projector Pn,J and Pm,S belongs to L1 (H) and have the decomposition S A.10.(ii). The F-linear span of n,m range(Pn,J Pm,S ) is dense in H. In particular, for each x ∈ range(Pn,J Pm,S ) we have J k Sl x = λkJ λlS πnk+ml Pn,J Pm,S x. Applying § A.9 to commuting operators J k and Sl for each k, l ∈ N and using the base of H we get projectors Pn,J and Pm,S which commute for each n and m, consequently, JS = UJUS λJ λS ∑n≥0,m≥0 πn+m Pn,J Pm,S , hence UJS = UJ US , λJS = λJ λS , Pl,JS = ∑n+m=l Pn,J Pm,S . In view of the H¨older inequality v 1/v ≤ ν (J)ν (S). νv (JS) = inf(∑∞ q r n=0 sn,JS dimF Pn,JS (H)) A.13. Proposition. If E is the normed space and H is the Banach space over the field F (complete relative to its uniformity), then Lr (E, H) is the Banach space such that if J, S ∈ Lr (E, H), then kJ + Skr ≤ kJkr + kSkr ;

kbJkr = |b| kJkr for each b ∈ K;

kJkr = 0 if and only if J = 0, where 1 ≤ r ≤ ∞, k ∗ kq := νq (∗). Proof. In view of Proposition A.11 it remains to prove that Lr (E, H) is complete, when H is complete. Let {Tα } be a Cauchy net in Lr (E, H), then there exists T ∈ L(E, H) such that limα Tα x = T x for each x ∈ E, since Lr (E, H) ⊂ L(E, H) and L(E, H) is complete. We demonstrate that T ∈ Lr (E, H) and Tα converges to T relative to νr for 1 ≤ r < ∞.

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Let αk be a monotone subsequence in {α} such that νrr (Tα − Tβ ) < 2−k−2 for each α, β ≥ αk , where k ∈ N. Since Tαk+1 − Tαk ∈ Lr (E, H), then (Tαk+1 − Tαk )x = ∑∞ n=1 an,k (x)yn,k k+p−1 ∞ ∞ r r −k−2 with ∑n=1 kan,k k kyn,k k < 2 . Therefore, (Tαk+p − Tαk )x = ∑h=k ∑n=1 an,h (x)yn,h for each p ∈ N, consequently, using convergence while p tends to ∞ we get (T − Tαk )x = ∞ ∞ ∞ r r r −k−1 , hence T − ∑∞ h=k ∑n=1 an,h (x)yn,h . Then νr (T − Tαk ) ≤ ∑h=k ∑n=1 kan,h k kyn,h k ≤ 2 Tαk ∈ Lr (E, H) and inevitably T ∈ Lr (E, H). Moreover, νr (T −Tα ) ≤ νr (T −Tαk )+νr (Tαk − Tα ) ≤ 2−(k−1)/r 2 for each α ≥ αk . A.14. Proposition. Let E, H, G be normed spaces over spherically complete F. If T ∈ L(E, H) and S ∈ Lr (H, G), then ST ∈ Lr (E, G) and νr (ST ) ≤ νr (S)kT k. If T ∈ Lr (E, H) and S ∈ L(H, G), then ST ∈ Lr (E, G) and νr (ST ) ≤ kSkνr (T ). Proof. For each δ > 0 there are bn ∈ H ∗ and zn ∈ G such that Sy = ∑∞ n=1 bn (y)zn for ∞ ∞ r r r ∗ each y ∈ H and ∑n=1 kbn k kzn k ≤ νr (S) + δ. Therefore, ST x = ∑n=1 T bn (x)zn for each ∗ r r r ∗ x ∈ E, hence νr (ST ) ≤ ∑∞ n=1 kT bn k kzn k ≤ kT k[νr (S) + δ], since kT bn (x)k = |bn (T x)| ≤ ∗ ∗ ∗ kbn kkT xk ≤ kbn kkT kkxk, where T ∈ L(H , E ) is the adjoint operator such that b(T x) =: (T ∗ b)(x) for each b ∈ H ∗ and x ∈ E. The operator T ∗ exists due to the Hahn-Banach theorem for normed spaces over the spherically complete field F [NB85, Roo78]. A.15. Proposition. If T ∈ Lr (E, H), then T ∗ ∈ Lr (H ∗ , E ∗ ) and νr (T ∗ ) ≤ νr (T ), where E and H are over the spherically complete field F. Proof. For each δ > 0 there are an ∈ E ∗ and yn ∈ H such that T x = ∑∞ n=1 an (x)yn for ∞ r r r ∗ each x ∈ E and ∑n=1 kan k kyn k ≤ νr (T ) + δ. Since (T b)(x) = b(T x) = ∑∞ n=1 an (x)b(yn ) ∗ (b)a , where y∗ (b) := b(y ), that is correct for each b ∈ H ∗ and x ∈ E, then T ∗ b = ∑∞ y n n n n=1 n due to the Hahn-Banach theorem for E and H over the spherically complete field F [NB85, r r r ∗ Roo78]. Therefore, νrr (T ∗ ) ≤ ∑∞ n=1 kyn k kan k ≤ νr (T ) + δ, since ky kH ∗ = kykH for each y ∈ H. A.16. Comments. Bounded operators on non-Archimedean Banach spaces were investigated, for example, in [Gru66, Put68, Roo78, LD02, Lud0341] and references therein. In this book only some specific results in operator theory are used, so there are not referred all works on this subject.

Appendix B

Non-Archimedean Polyhedral Expansions B.1.

Ultra-uniform Spaces

B.1.1. Let us recall that by an ultra-metric space (X, ρ) is implied a set X with a metric ρ such that it satisfies the ultra-metric inequality: ρ(x, y) ≤ max(ρ(x, z); ρ(z, y)) for each x, y and z ∈ X. A uniform space X with ultra-uniformity U is called an ultra-uniform space (X,U) such that U satisfies the following condition: |x − z| < V ′ , if |y − z| < V ′ and |x − y| < V , where V ⊂ V ′ ∈ U, x, y and z ∈ X [Eng86, Roo78]. If in X a family of pseudo-ultra-metrics P is given and it satisfies conditions (UP1,UP2), then it induces an ultra-uniformity U due to proposition 8.1.18 [Eng86]. Let L be a non-Archimedean field. We say that X is a L-Tychonoff space, if X is a T1 -space and for each F = F¯ ⊂ X with x ∈ / F there exists a continuous function f : X → B(L, 0, 1) such that f (x) = 0, f (F) = {1}, where B(X, y, r) := {z ∈ X : ρ(y, z) ≤ r} for y ∈ X and r ≥ 0. From Ind(L) = 0 it follows Ind(X) = 0 (see §6.2 and Chapter 7 in [Eng86]). Since the norm | ∗ |L : L → Γ˜ L is continuous, then X is the Tychonoff space, where Γ˜ L := {|x|L : x ∈ L} ⊂ [0, ∞). Vice versa if X is a Tychonoff space with Ind(X) = 0, then it is also L-Tychonoff, since there exists a clopen (closed and open at the same time) neighborhood W ∋ x with W ∩ F = 0/ and as the locally constant function f may be taken with f (x) = 0 and f (W ) = {1}. Let us consider spaces C(X, L) := { f : X → L| f is continuous } and C∗ (X, L) := { f ∈ C(X, L) : | f (X)|L is bounded in R}, then for each finite family { f1 , . . . , fm } ⊂ C(X, L) (or C∗ (X, L)) the following pseudo-ultra-metric is defined: ρ f1 ,..., fm (x, y) := max(| f j (x) − f j (y)|L : j = 1, . . . , m}. Families P or P∗ of such ρ f1 ,..., fm induce ultra-uniformities C or C∗ respectively and the initial topology on X. If a sequence {V j : j = 0, 1, . . . } ⊂ U is such that V0 = X 2 , pV j+1 ⊂ V j for j = 1, 2, . . . , where p is a prime number, then there exists a T pseudo-ultra-metric ρ(x, y) := 0 for (x, y) ∈ ∞j=0 V j , ρ(x, y) = p− j for (x, y) ∈ V j \V j+1 , so Vi ⊂ {(x, y) : ρ(x, y) ≤ p−i } ⊂ Vi−1 . Indeed, from (x, y) ∈ Vi \Vi+1 and (y, z) ∈ V j \V j+1 for j ≥ i it follows (x, z) ∈ Vi and ρ(x, z) ≤ p−i = ρ(x, y). Therefore, ultra-uniform spaces may be equivalently characterized by U or P (see § 8.1.11 and § 8.1.14 in [Eng86] ). Henceforth, locally compact non-discrete non-Archimedean infinite fields L are con-

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sidered. If the characteristic char(L) = 0, then due to [Wei73] for each such L there exists a prime number p such that L is a finite algebraic extension of the field Qp of p-adic numbers. If char(L) = p > 0, then L is isomorphic with the field Fpn (θ) of the formal power series by the indeterminate θ, where n ∈ N, p is the prime number, each z ∈ L has the j −1 up to the equivalence of the n form z = ∑+∞ j=k(z) a j θ with a j ∈ Fp for each j, |θ|L = p non-Archimedean normalization, k(z) ∈ Z, Fpn is the finite field consisting of pn elements, Fpn ֒→ L is the natural embedding. For an ordinal A with its cardinality m = card(A) by c0 (L, A) it is denoted the following Banach space with vectors x = (xa : a ∈ A, xa ∈ L) of a finite norm kxk := supa∈A |xa |L and such that for each b > 0 a set {a ∈ A : |xa |L ≥ b} is finite. It has the orthonormal in the non-Archimedean sense basis {e j := (δ j,a : a ∈ A) : j ∈ A}, where δ j,a = 1 for j = a and δ j,a = 0 for each j 6= a [Roo78]. If card(A1 ) = card(A2 ) then c0 (L, A1 ) is isomorphic with c0 (L, A2 ). In particular c0 (L, n) = Ln for n ∈ N. Then card(A) is called the dimension of c0 (L, A), card(A) = dimL c0 (L, A). B.1.2. Lemma. Let (X, ρ) be an ultra-metric space, then there exists an ultra-metric ρ′ equivalent to ρ such that ρ′ (X, X) ⊂ Γ˜ L . Proof. Let ρ′ (x, y) := supb∈Γ˜ L ,b≤ρ(x,y) b, where x and y ∈ X, either L ⊃ Qp or L = Fpn (θ). Then ρ′ is the ultra-metric such that ρ′ (x, y) ≤ ρ(x, y) ≤ p × ρ′ (x, y) for each x and y ∈ X. B.1.3. Lemma. Let (X, P) be an ultra-uniform space, then there exists a family P′ such that ρ′ (X, X) ⊂ Γ˜ L for each ρ′ ∈ P′ ; (X, P) and (X, P′ ) are uniformly isomorphic, the completeness of one of them is equivalent to completeness of another. Proof. In view of Lemma 2 for each ρ ∈ P there exists an equivalent pseudo-ultrametric ρ′ . They form a family P′ . Evidently, the identity mapping id : (X, P) → (X, P′ ) is the uniform isomorphism. The last statement follows from 8.3.20 [Eng86]. B.1.4. Theorem. For each ultra-uniform space (X, ρ) there exist an embedding f : X → B(c0 (L, AX ), 0, 1) and an uniformly continuous embedding into c0 (L, AX ), where card(AX ) = w(X), w(X) is the topological weight of X. Proof. In view of theorem 7.3.15 [Eng86] there exists an embedding of X into the Baire space B(m), where m = w(X) ≥ ℵ0 . In the case w(X) < ℵ0 this statement is evident, since X is finite. In view of lemma 2.2 we choose in B(m) an ultra-metric ρ equivalent to the initial one with values in Γ˜ L such that ρ({xi }, {yi }) = p−k , if xk 6= yk and xi = yi for i < k, ρ({xi }, {yi }) = 0, if xi = yi for all i, where {xi } ∈ B(m), i ∈ N, xi ∈ D(m), D(m) denotes the discrete space of cardinality m. Let A = N × C, card(C) = m, {ei,a |(i, a) ∈ A} be the orthonormal basis in c0 (L, A). For each {xi } ∈ B(m) we have xi ∈ D(m) and we can take xi = ei,a for suitable a = a(i), since D(m) is isomorphic with {ei,a : a ∈ C). Let f ({xi }) := ∑i∈N,a=a(i) pi ei,a , consequently, k f ({xi }) − f ({yi })kc0 (L,A) = ρ({xi }, {yi }). The last statement of the theorem follows from the isometrical embedding of (X, ρ) into the corresponding free Banach space, which is isomorphic with c0 (L, A) (see theorem 5 in [Lud95] and Theorems 5.13 and 5.16 in [Roo78] ). For each ultra-metric ρ ∈ P of an ultra-uniform space (X, P) there exists the equivalence relation Rρ such that xRρ y if and only if ρ(x, y) = 0. Then there exists the quotient mapping gρ : X → Xρ := (X/Rρ ) , where Xρ is the ultra-metric space with the ultra-metric also denoted by ρ, X˜ denotes the completion of X. Then X has the uniform embedding into ˜ Hence we have got the following. the limit of the inverse spectra inv − limρ Xρ = X. B.1.5. Corollary. Each ultra-uniform space (X, P) has a topological embedding into

Non-Archimedean Polyhedral Expansions

181

∏ρ∈P B(c0 (L, Aρ ), 0, 1) and an uniform embedding into ∏ρ∈P c0 (L, Aρ ) with card(Aρ ) = w(Xρ . B.1.6. Corollary. For each j ∈ N and f (X) ⊂ c0 (L, AX ) from Theorem 4 there are coverings U j of the space f (X) by disjoint clopen balls Bl with diameters not greater than p− j and with infl6=k dist(Bl , Bk ) > 0. Proof follows from the consideration of the covering of c0 (L, A) by balls B(c0 (L, A), x, r) with 0 < r ≤ p− j and x ∈ c0 , since such balls are either disjoint or one of them is contained in another and ΓL is discrete in (0, ∞), where ΓL := Γ˜ L \ {0}. Then S −j q∈J B(c0 , xq , rq ) is the clopen ball in c0 with r ≤ p , if all balls in the family J have nonvoid pairwise intersections. Taking B(c0 , x, r) ∩ f (X) we get the statement for f (X) using the transfinite sequence of the covering. B.1.7. Note. A simplex s in Rn may be taken with the help of linear functionals, for example, {e j : j = 0, . . . , n}, where e j = (0, . . . , 0, 1, 0, . . . , 0) with 1 in the j-th place for j > 0 and e0 = e1 + · · · + en , s := {x ∈ Rn : e j (x) ∈ [0, 1] for j = 0, 1, . . . , n}. In the case of Ln , if to take B(L, 0, 1) instead of [0, 1], then conditions x j := e j (x) ∈ B(L, 0, 1) for j = 1, . . . , n imply e0 (x) = x1 + · · · + xn ∈ B(L, 0, 1) due to the ultra-metric inequality ( since B(L, 01, ) is the additive group ), that is s = B(L, 0, 1). Moreover, its topological border is empty Fr(s) = 0/ and Ind(Fr(s)) = −1. Let us denote by πL an element from L such that B(L, 0, 1− ) := {x ∈ L : |x|L < 1} = πL B(L, 0, 1) and |πL |L = supb∈ΓL ,b 0, where dist(s, q) := infx∈s,y∈q ρ(x, y). By vertices of the simplex s = B(c0 (L, A), 0, 1) we call points x = (x j ) ∈ c0 (L, A) such that x j = 0 or x j = 1 for each j ∈ A, dimL (s) := card(A). For each E ⊂ A, E 6= A and a vertex e by verge of the simplex s we call a subset e + B(c0 (L, E), 0, 1) ⊂ s. For an arbitrary simplex its verges and vertices are defined with the help of affine transformation as images of verges and vertices of the unit simplex B(c0 (L, A′ ), 0, 1). Then in analogy with the classical case there are naturally defined notions of a simplicial complex K and his space |K|, also a sub-complex and a simplicial mapping. The latter has restrictions on each simplex of polyhedra that are affine mappings over the field L and images of vertices are vertices. Instead of the barycentric subdivision in the classical case we introduce a p j -subdivision of simplexes and polyhedra for j ∈ N and L ⊃ Qp , that is a partition of each simplex B(c0 (L, A′ ), x, r) into the disjoint union of simplexes with diameters equal to rp− j . Each simplex s with dimL s = card(A′ ) may be considered also in c0 (L, A), where A′ ⊂ A, since there exists the isometrical embedding c0 (L, A′ ) ֒→ c0 (L, A) and the projector π : c0 (L, A) → c0 (L, A′ ). By a dimension of a polyhedron P we call dimL P := sup(s⊂P,s is a simplex ) dimL s. The polyhedron P is called locally finite-dimensional if all simplexes s ⊂ P are finite dimensional over L, that is, dimL s ∈ N. For a simplex s = B(c0 (L, A′ ), x, r) by L-border ∂s we call the union of all its verges q with the codimension over L equal to 1 in c0 (L, A′ ) =: X, that is, q = e + B′ , where B′ are balls in

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c0 (L, A”) =: Y , (A′ \ A”) is a singleton, A” ⊂ A′ , X ⊖ Y = L, Y ֒→ X. For the polyhedron S S P = j∈F s j by the L-border we call ∂P = j∈F ∂s j , where F is the set. (2). A continuous mapping f of a set M, M ⊂ c0 (L, A) into a polyhedron P we call essential, if there is not any continuous mapping g : M → P for which are satisfied the following conditions: (i) g(M) does not contain P; (ii) there exists M0 ⊂ M, M0 6= M with f (M) ∩ ∂P = f (M0 ) = g(M0 ) ⊂ ∂P and their restrictions coincide f |M0 = g|M0 ; (iii) if f is linear on [x, y] := {tx + (1 − t)y|t ∈ B(L, 0, 1)} ⊂ M, then g is also linear on [x, y] such that g(x) 6= g(y) for each f (x) 6= f (y). (3). The function f from § 2.8.(2) is inessential, if there exists such g. (4). Let f : M → N be a continuous mapping, c0 (L, A) ⊃ N ⊃ P, P be a polyhedron. Then P is called essentially (or unessentially) covered by N under the mapping f , if f | f −1 (P) is essential (or inessential respectively). (5). Let f : M → P and g : M → P are continuous mappings, where M is a set, P is a polyhedron. Then g is called a permissible modification of f , if three conditions are satisfied: (i) from a ∈ M and f (a) ∈ s it follows g(a) ∈ s, where s is a simplex from P; (ii) if x and y ∈ M, [x, y] ⊂ M and f : [x, y] → P is linear, then g : [x, y] → P is also linear (over L) and g(x) 6= g(y) for each f (x) 6= f (y); (iii) f (∂M) = g(∂M). (6). The mapping f : M → P is called reducible (or irreducible), when it may (or not respectively) have the permissible modification g such that f (M) is not the subset of g(M). (7). A mapping f : P → Q for polyhedra P and Q is called normal, if (i) ρQ ( f (x), f (y)) ≤ ρP (x, y) for each x and y ∈ P, that is f is a non-stretching mapping; (ii) there exists a p j -subdivision Q′ of polyhedra Q such that f : P → Q′ is a simplicial mapping, that is, f |s is affine on each simplex s ⊂ P and f (s) is a simplex from Q′ . j (8). Let X = inv − lim j {X j , fi , E} be an expansion of X into the limit of inverse spectra of polyhedra X j over L. This expansion is called (a) irreducible, if for each open V ⊂ X j there exists a co-final subset EV ⊂ E such that {X j , fi , EV } is also the irreducible polyhedral j representation of the space V , that is fi : X j → Xi are irreducible and surjective for each j i ≥ j ∈ EV . The polyhedral system (representation) {X j , fi , E} is called (b) n-dimensional, if dimL X j ≤ n for each j ∈ E, sup j∈E dimL X j = n. j B.1.9. Notes. Conditions 2.8.(2(iii), (5(ii, iii)) and restrictions on fi in 8.(8(a)) are additional in comparison with the classical case. They are imposed in § 8, since there exists a continuous non-linear retraction r : s j → ∂s j for the non-Archimedean field L, which may be constructed with the help of p-subdivision. If f is simplicial on each polyhedron M and 2.8.(2(iii)) is accomplished, then dimL g(M) = dimL f (M). Here we mean by ANRU an ultra-uniform space X such that under embedding into an ultra-uniform space Y it is the uniformly continuous retract r : V → X of its uniform neighborhood V , X ⊂ V ⊂ Y . We denote by U(X,Y ) for two ultra-uniform spaces X and Y an ultra-uniform space of uniformly continuous mappings f : X → Y with the uniformity generated by a base of the form W = {( f , g)|( f (x), g(x)) ∈ V for each x ∈ X}, where V ∈ V, V is a uniformity on Y corresponding to PY .

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Here we call an ultra-uniform space Y injective, if for each ultra-uniform space X with H ⊂ X and an uniformly continuous mapping f : H → Y there exists an uniformly continuous extension f : X → Y . B.1.10. Theorem. B(c0 (L, A), 0, 1) is an injective space for card(A) < ℵ0 . Proof may be done analogously to theorem 9 on p. 40 in [Isb64]. B.1.11. Theorem. Each uniform polyhedron P over L is ANRU. Proof. Since a = infi6= j∈F dist(si , s j ) > 0, b = supi∈F diam(si ) < ∞, then there exists a uniform covering U such that for each si there exists a/p-clopen neighborhood Vi ∈ U. Each si is an uniform retract Vi , ri : Vi → si , consequently, there exists uniformly continuous S retraction r : S = Vi ∈U Vi → P such that r|si = ri for each i, since supi∈F diam(si ) < ∞. The neighborhood S is uniform, since St(M,U) = S. B.1.12. Note. Further for uniform spaces are considered uniformly continuous mappings and uniform polyhedra and for topological spaces continuous mappings and polyhedra if it is not specially outlined. B.1.13. Lemma. Let (X, P) be an ultra-uniform strongly zero-dimensional space, P be a polyhedra over L, A1 ,. . . ,Aq are non-intersecting closed subsets in X, q ∈ N, card(P) ≥ q. Then there exists an uniformly continuous mapping f : X → P such that f (Ai ) ∩ f (A j ) = 0/ for each i 6= j. Proof. There exists a disjoint clopen covering V j for X satisfying A j ⊂ V j for each j = 1, . . . , q (see Theorem 6.2.4 [Eng86]). Then we can take locally constant mapping f with f (V j ) ⊂ s j ⊂ P. B.1.14. Lemma. Each non-stretching mapping f : E → P has non-stretching continu˜ ρ), where P is a uniform polyhedron over L, E ⊂ X. ation on (X, Proof. There exists an embedding P ֒→ c0 (L, A) for card(A) = w(P) due to Lemma 4 [Lud95]. We choose f : E → c0 (L, A) with f = ( f i : i ∈ A), f i : E → Lei , where (ei : i ∈ A) is the orthonormal basis in c0 (L, A) and inf(i6= j,si and s j ⊂P) dist(si , s j ) > 0, si are simplexes from P. B.1.15. Definition. An ultra-uniform space (X, P) is called LE-space, if each uniformly continuous f : Y → L has an uniformly continuous extension on all X, where Y ⊂ X. B.1.16. Theorem. An ultra-metric space X is an LE-space if and only if X˜ = inv − limm {Xm , fnm , E}, where Xm are fine spaces and bonding mappings fnm : Xm → Xn are uniformly continuous for each m ≥ n ∈ E. Proof. Let us consider XA = B(c0 (L, A), 0, 1) for card(A) ≥ ℵ0 or XA = Ln which are not fine spaces, since on XA there are continuous f : XA → L which are not uniformly continuous. Then for the embedding of the non-fine space XA ֒→ c0 (L, A) there is not compact H with XA ⊂ H ⊂ c0 (L, A). Therefore, in XA there exists a countable closed subset of isolated points Y = {xi : i ∈ N) and a continuous function f : Y → L with | f (x j ) − f (xi )|/|x j − xi | > ci > 0 for each j ≥ i and limi→∞ ci = ∞ such that f has a continuous extension g on XA . Indeed, there exists a clopen neighborhood W , that is a b-enlargement (with b > 0) relative to the ultra-metric ρA on XA , so there is a continuous retraction r : W → Y , g(x) = const ∈ L on (XA \W ), g(x) = f (r(x)) on W , g|Y = f . Therefore, g is continuous and uniformly continuous on XA (see also Theorem 1.7 [CI60]).

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B.2.1. Theorem. Each complete ultrauniform space (Y, P) is a limit of an inverse spectra of ANRU Y j , where Y j are embedded into complete locally L-convex spaces. Proof. In view of Corollary 1.5 there exists a uniform embedding Y ֒→ ∏ρ∈P c0 (L, Aρ ) =: H. In each c0 (L, Aρ ) may be taken the orthonormal basis {e j,ρ : j ∈ Aρ }, card(Aρ ) = w(Yρ ) and define canonical neighborhoods U( f , b; ( j1 , ρ1 ), . . . , ( jn , ρn )) := {q ∈ H : |π ji ,ρi (q) − π ji ,ρi ( f )| < b for i = 1, . . . , n}, where π j,ρ : H → Le j,ρ are projectors, f ∈ H, b > 0, n ∈ N. Each clopen subset Zb := H \ U( f , b; ( j1 , ρ1 ), . . . , ( jn , ρn )) is uniformly continuous (non-stretching) retract of Zb/p , Tq that is Zb is ANRU. Analogously each finite intersections l=1 Z( fl , bl , ( j1l , ρl1 ), . . . , T T ( jnl , ρln )) = kl=1 Zl,bl are also ANRU, since Zk,bk ⊃ kl=1 Zl,bl and a retraction gk : Zk,bk /p → Zk,bk produces non-stretching relative to the corresponding pseudo-ultra-metric retraction T T Tq Tq gk : kl=1 Zl,bl /p → kl=1 Zl,bl . Hence gq ◦ · · · ◦ g1 = g : l=1 Zl,bl /p → l=1 Zl,bl is the Tq (non-stretching) retraction. Let S be the ordered family of such l=1 Zl,bl ⊂ H \ Y , then T Z∈S (H \ Z) = Y . Further as in the proof of Theorem 7.1 [Isb61]. B.2.2. Lemma. Let (X, ρX ) and (Y, ρY ) be ultra-metric spaces, f : X → Y be a continuous mapping such that f |H is a b-mapping, where H is dense in X, b > 0. Then X and Y may be embedded into a Banach space W over L such that X with equivalent ultra-metric in W and k f (x) − xk ≤ b for each x ∈ X, that is, f is a b-shift or a b-mapping. Proof. From Theorem 1.4 and Lemma 1.3 and Corollary 1.6 it follows that there exists such embedding of X into the corresponding W = c0 (L, A) with card(A) = w(X) with the disjoint clopen covering V = {B(c0 (L, A), x j , b j ) =: B j : x j ∈ H, ∞ > b j > 0, j ∈ F}. Let Y j := f (B j ) ⊂ Y , consequently, diam(Y j ) ≤ b, since f is the b-mapping, that is f realizes the covering of X consisting from subsets of diameters not greater than b. Then ρX (x′ , x”) ≤ max(ρ(x′ , x j ), ρ(x j , x”)), where x” ∈ f −1 (y), y ∈ Y j , x′ ∈ B(c0 (L, A), x j , b j ), f (x′ ) = y, diam( f −1 (y)) ≤ b. Let xi 6= x j , this is equivalent to Bi ∩ B j = 0/ and is equivalent S to ρX (xi , x j ) > b, consequently, Yi ∩Y j = 0/ and Y = j∈F Y j . In view of continuity of f there exists the embedding of Y j into B j , since f is the b-mapping, w(Y ) ≤ w(X), 0 < b j ≤ b and due to theorem 2.4. B.2.3. Lemma. Let there exists a non-stretching (uniformly continuous) mapping f : R → P and a non-stretching (uniformly continuous) permissible modification g : M → P, where R is a complete ultra-metric space (LE-space respectively), P is a polyhedron, M is a subspace in R; if R is a polyhedron let M be a sub-polyhedron. Then g has a nonstretching (uniformly continuous respectively) extension on the entire R and this extension is a permissible modification of f . Proof. For a complete ultra-metric space R the mapping g has the non-stretching (uniformly continuous) extension on the completion of M which coincides with the closure M¯ of M in R, since R and P are complete. The space P is complete, since it is ANRU by Theorem 1.11 (see also Theorem 1.7 [Isb59], Theorems 8.3.6 and 8.3.10 [Eng86]). For the embedding R ֒→ c0 (L, A) with card(A) = w(R) by Theorem 1.4 it may be taken due to Corollary 1.6 the disjoint clopen covering V such that each W ∈ V has the form W = R ∩ B(c0 (L, A), x, rW ), where rW > 0, x ∈ R, supW ∈V rW ≤ p− j (for each j ∈ N there exists such V ). In view of uniform continuity of f and uniformity of the polyhedron P there exists V

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such that for each W from V there exists a simplex T ⊂ P with f (W ) ⊂ T . The space c0 (L, A) has the orthonormal basis {e j : j ∈ A}. If f is linear on no any [a, b] ⊂ R, then for the construction of g may be used lemma 2.14 or Theorem 1.16. This may be done with the help of transfinite induction by the cardinality of sets of vertices of simplices from P (or using the Teichm¨uller-Tukey lemma) and with the modification of the proof of Lemmas 1 ¯ contains each zero-dimensional simplex T 0 ⊂ P. If it in [Fre37, Isb61]. In general let g(M) is not so, then there exists a point b = f −1 (T 0 ) in which g is not defined. If f is non-linear ¯ then g(b) = f (b). If f is linear on such [a, b], then from on each [a, b] ⊂ R with a ∈ M, ρ(a, b) < ∞ it follows that [a, b] is homeomorphic with the compact ball B(L, a, ρ(b, a)) in L and f ([a, b]) is homeomorphic with B(L, f (a), k f (b) − f (a)k). Then in M¯ ∩ [a, b] = Y there exists y such that ρ(y, a) = supx∈Y ρ(a, x) = t. The subspace Y is covered by the finite number of W ∈ V . For y 6= a, that is t > 0, f ([a, b]) is compact and is contained in the finite number of simplexes from P, consequently, there exists the permissible modification of g on [a, b] also. Let E and F be two subsets in R. We denote by sp(E, F, f ) the subspace S cl(( (a∈E,b∈F, f |[a,b] is L− linear ) [a, b]) ∪ E ∪ F) in R, where cl(S) denotes the closure of

S in R for S ⊂ R. If B = {q : f −1 (T 0 ) = q, T 0 ⊂ P, g is not defined in q}, then by the ¯ B, f ) =: M0 . Let Teichm¨uller-Tukey lemma there exists the extension of g on sp(M, S S M j := sp( i< j Mi , B, f ), where B = (T j is not the subset of g(M)) T j , T j are simplexes from P with the cardinality of sets of vertices equal to j, where j ≤ w(P). From Lemma 2.2 it follows that conditions 1.8.(5(i − iii)) may be satisfied on M j ∩ R. Considering verS tices of s from R ∩ f −1 (T j ) \ i< j Mi we construct g on M j ⊂ R. Since R ֒→ c0 (L, A), then sup j M j = R, where M j are ordered by inclusion: M j ⊃ Mi for each i ≤ j. Henceforth, we assume that for uniformly continuous mapping f : Y → P are satisfied Conditions 1.14 and 1.16, where Y ⊂ (X, ρ). B.2.4. Lemma. Let f : M → P is irreducible, M and P are polyhedrons, N is a subpolyhedron in M, Q is a sub-polyhedron in P, f (N) ⊂ Q, then a mapping f : N → Q is irreducible. Proof. Let f : N → Q be reducible, that is there exists an permissible modification g : N → Q with g(N) not contained in f (N), q ∈ f (N) \ g(N). In view of Lemma 2.3 there exists the extension g : M → P. Let r > 0 is sufficiently small and U := Nr = {y ∈ M : there exists x ∈ N with ρ(x, y) ≤ r} be the r-enlargement of the subspace N such that q ∈ / g(Nr ). Since M and P are (uniform) polyhedra and f is (uniformly) continuous and U is clopen in M, then there exists p j -subdivision M ′ and clopen polyhedron H with Nr/p ⊂ N ⊂ U ⊂ M ′ such that h|H = g|H , M ′ \ H is the sub-polyhedron in M ′ , h|M\H = f |M\H . Then q∈ / h(H) and from the irreducubility of f it follows that q ∈ / f (M \ H), consequently, h is the permissible modification of f and h(M) is not the subset of f (M). This contradiction lead to the statement of this lemma. B.2.5. Lemma. Let f : M → P, M and P be polyhedrons over L. Then the condition of irreducibility of f is equivalent to that each sub-polyhedron in Q ⊂ P is essentially covered. Proof. If f is irreducible, then due to Lemma 2.4 each sub-polyhedron Q is essentially covered. Let vice versa each Q is essentially covered and f has the permissible modification g with P = f (m) not contained in g(M). From f (∂M) = g(∂M) and that f is essential the contradiction with {P is not contained in g(M)} follows, consequently, f is irreducible. B.2.6. Lemma. Let P and M be polyhedrons. If f : M → P has only admissible modifi-

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cations, then f is irreducible. Proof. From f (∂M) = g(∂M) and that g is the permissible modification of f it follows that each sub-polyhedron Q from P is essentially covered due to lemma 3.3. In view of Lemma 2.5 f is irreducible. B.2.7. Lemma. Let there is an inverse spectra S = {Rm , fnm , E} of polyhedra Rm over L, fnm are simplicial mappings, gl : Rl → P, gn = gl ◦ fln for a fixed l, fn = inv − limm fnm , g = gl ◦ fl , R = inv − lim S, fn : R → Rn , E is linearly ordered. If g : R → P is reducible for a polyhedron P, then for almost all n (that is, there exists k ∈ E such that for each n ≥ k) gn : Rn → P are reducible. B.2.8. Lemma. If f : M → N, g : N → T , f (M) = N, where M and N are polyhedra, then from g is inessential (reducible) it follows f ◦ g is inessential (reducible). B.2.9. Lemma. Suppose that there is given an irreducible inverse mapping system S = {Pm , fnm , E} of polyhedra Pm over L, M is closed in P = inv − lim S such that Mn := fn (M) m are sub-polyhedra in Pn and for each m > l permissible modifications gm l for f l are given m n (on the entire Pm ) and for each n > m mappings gl ◦ fm are permissible modifications of n flm . Then the inverse mapping system SM := {Mn , gm l ◦ f m , El } is irreducible, where El = {n ∈ E : n ≥ l}. Proof is analogous to the proofs of Lemmas IV.30.V-VII in [Fre37] using the preceding lemmas (see also [Isb61]). Indeed, in Lemma 2.8, if f is L-linear on [a, b] ⊂ M and g is L-linear on [ f (a), f (b)], then f ◦ g is L-linear on [a, b]. In Lemma 2.9 from surjectivity and n irreducibility of flm it follows surjectivity and irreducibility of gm l ◦ f m due to Lemmas 2.4, 2.5 and 2.8. Since El is cof-inal with E, then for each W open in M there exists V open in P such that W = M ∩V . B.2.10. Lemma. If T is a simplex from a p j -subdivision P′ of polyhedron P then for each clopen neighborhood U ⊃ T such that U is a sub-polyhedron in P′ there exists a mapping k : P → P such that k|U is simplicial and k(U) = T . Proof. In view of Theorem 1.11 there exists the retraction r : P → U (it is uniform if P is the uniform polyhedron), then analogously to the proof of Lemma VIII [Fre37]. B.2.11. Lemma. Let P be a polyhedron, f : M → P and g : M → P are two b-close mappings (that is, ρ( f (a), f (b)) ≤ b for each a ∈ M), then there exists k from lemma 3.10 such that kg and f are b-close and kg is a permissible modification of f . B.2.12. Lemma. Let f : P → Q be a non-stretching mapping of a uniform polyhedron P onto a uniform polyhedron Q over L. Then there exists a p j -subdivision P′ of P and a normal mapping g : P → Q such that g is a permissible modification of f . Proof. For each b > 0 in view of uniform continuity of f there exists p j -subdivision ′ P of P and pi -subdivision of Q and a simplicial mapping h : P′ → Q, which is b-close to f . Indeed, simplexes T l ⊂ P′ are disjoint and clopen for them due to Lemmas 3 and 4 [Lud95] h can be chosen such that: (i) |h(e) − f (e)| < b for linearly independent vertices el, j ∈ T l , that is, {(el, j − ei,0 ) : j ∈ Al } are linearly independent, el,0 is a marked vertex from T l , card(Al ) = dimL T l and (ii) h(T l ) are simplexes in Q with diam(h(T l )) ≤ diam(T l ) for suitable b > 0 and pi -subdivision Q′ of Q, where h|T l are affine mappings for each l. This is possible due to uniformity of polyhedra P and Q. Taking g = k ◦ h, where k is the suitable mapping from Lemma 2.11 we get the desired g. B.2.13. Lemma. Let g be a permissible modification of f : R → P and h : P → Q is a normal mapping, where P and Q are uniform polyhedra. Then h ◦ g is the permissible

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modification of h ◦ f . B.2.14. Lemma. Let {Pn , fmn , E} = S be an irreducible inverse system of uniform polyhedrons over L, M be closed in P = inv − lim S, Mn = fn (M) be sub-polyhedra in Pn , {Qk , gkl , F} be an inverse spectra of polyhedra Qk , which appear by p j(k) -subdivisions n(k) of Pn(k) , gkl be normal and permissible modifications of fn(l) , where F is co-final with E, Nk = gk (M), gkl |Nk and gl |M are irreducible. Then Nk are polyhedra. Proof follows from Lemmas 3.7-3.12 (see also [Fre37, Isb61]). B.2.15. Lemma. Suppose that for a complete ultra-metric space R there is a nonstretching mapping f : R → P, f (R) = P, P is a uniform polyhedron over L. Then for each b > 0 there exists a b-mapping g : R → Q and Q is a uniform polyhedron over L such that for sufficiently fine p j -subdivision Q′ of a polyhedron Q there exists a normal mapping h : Q → P and h ◦ g is a non-stretching permissible modification of f . Proof. For R there exists the embedding R ֒→ c0 (L, A) with card(A) = w(R) by theorem 2.4 and a clopen neighborhood S with R ⊂ S ⊂ R(r) that is a uniform polyhedron due to Corollary 1.6, where R(r) denotes the r = b/p-enlargement of R. By Lemma 2.2 there exists the sub-polyhedron Q with R ⊂ Q ⊂ S and the b′ -mapping g : R → Q. If [a, z] ⊂ S and f |[a,z] is L-linear, then we can choose g such that it is linear on [a, z] and g(a) 6= g(z) when f (a) 6= f (z). From the completeness of R and lemma 2.14 it follows that f has the non-stretching extension f : S → P. Then there are g, Q and non-stretching h : Q → P for sufficiently small r > 0 and b′ > 0. For h due to Lemma 2.12 there exists the permissible modification and normal k : Q′ → P such that h ◦ g(R) = P, f (∂R) = h ◦ g(∂R) for ∂R 6= 0, that is, when R contains simplexes T with dimL T > 0. Such k, h and g may be constructed on each simplex T from Q′ and then on the entire space. B.2.16. Lemma. Suppose that R is a complete ultra-metric space, fn : R → Pn are nonstretching bn -mappings, Pn are uniform polyhedra over L, n ∈ E, E is an ordered set such that for each b > 0 there exists l ∈ E with 0 < bn < b for n > l. Then there exists a normal irreducible inverse mapping system S = {Qm , gnm , F}, F is co-final with E, inv − lim S = R, Qm are sub-polyhedra of p j(m) -subdivisions of Pn(m) and gm are permissible modifications of fn(m) , where gn = inv − limn gnm : R → Qm . Proof. In view of Lemmas 2.5, 2.6, 2.13 and 2.14 it is sufficient at first to construct S with non-stretching normal normal and surjective mappings gm n . This can be done due Lemma 2.15 with gm ◦ f being non-stretching permissible modifications of fn(l) and n(m) l m inv − limm gl ◦ fn(m) are non-stretching permissible modifications of fn(l) . B.2.17. Lemma. If the ultra-metric space (X, ρ) is isomorphic with inv − lim{(Xm , ρm ); fnm ; F} and the following conditions are satisfied: (1) for each m there are embeddings qm : Xm ֒→ (E, ρ) into a complete space (E, ρ); (2) fm : X → Xm are projections; (3) (Xm , ρm ) are ultra-metric spaces; (4) there is given a family {bm > 0 : m ∈ F}, bm ∈ ρ(X, X) and for each b > 0 the set {m : bm > b} is finite, tm := infρ(x,y)>bm ρ(qm (x), qm (y)), limm tm = 0, for each m > n and x and the inequality ρ(qm (x), qn ◦ fnm (x)) < tn is accomplished. Then the mappings qm ◦ fm converge uniformly to the embedding X ֒→ E. ˜ p ). Proof. In view of Lemma 1.2 we may suppose that ρ(X, X) and ρm (Xm , Xm ) ⊂ Γ(Q If x and y ∈ X and ρ(x, y) > bn , then ρ(qn ◦ fn (x), qn ◦ fn (y)) ≥ tn . From conditions (2, 4, 5)

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it follows the existence of k such that ρ(qm ◦ fm (x), qm ◦ fm (y)) ≥ tn for each m > k and ultra-metric ρ, consequently, q = inv − limm qm is the embedding, since ρ(q(x), q(y)) ≥ tn . B.2.18. Theorem. Let (X, P) be a complete ultra-metric space and L be a locally compact field, L ⊃ Qp . Then there exists an irreducible normal expansion of (X, P) into the limit of the inverse mapping system S = {Pn , fmn , E} of uniform polyhedra Pn over L, moreover, inv − lim S is isomorphic with (X, P), in particular for ultra-metric space (X, ρ) the system S is the inverse sequence. Proof. From Corollary 1.5 and Theorem 2.1 it follows the existence of the expansion of j (X, P) into the uniformly isomorphic limit of the inverse mapping system R = {Y j , fi , F} of j ANRU Y j with non-stretching fi , where Y j are the complete ultra-metric spaces. Each Y j is closed in the finite products of the spaces c0 (L, Ak ). From Lemmas 2.2-2.17 it follows the existence of the irreducible normal uniform polyhedral expansion for each Y j , moreover, j j using the permissible modifications of gi of fi and the same lemmas we can construct the system of the entire space (X, P) with the same properties. We consider further uniform coverings V corresponding to the uniform polyhedra S P = W ∈V W , which due to theorem 2.11 are ANRU. Let ρ be the ultra-metric in X and ˜ ρ(X, X) ⊂ Γ(L). We can associate with V a pk -nerve with k ∈ Z, that is, an abstract simplicial complex Nk vertices of which are elements from V . Its simplexes are the spans of (pulled on) vertices W j satisfying ρ(W j ,Wi ) ≤ pk b, where b = supW ∈V diam(W ) < ∞, each rib [W j ,Wi ] from s has the lenghts no less than t = infW j 6=Wi ∈V ρ(W j ,Wi ) > 0. Then from Nk ֒→ c0 (L, Ak ) with card(Ak ) = w(Nk ) it follows that each s is uniformly isomorphic with some ball B(c0 (L, A), 0, 1), where card(A) = m ≤ w(Nk ). With each V is associated the equivalence relation: xRy if and only if there exists W ∈ V such that x and y ∈ W . Since V is disjoint and clopen, then the quotient mapping f : X → X/R is defined. With each V is associated the partition of the unity { fW : W ∈ V }, fW : X → B(L, 0, 1), fW (x) = 1 for x ∈ W and fW (x) = 0 for x ∈ / W , { fW : W ∈ V } is subordinated to V . There are the canonical non-stretching mappings Fk : X → Nk . If X is compact, then X/R is the finite discrete space and dimL Nk = n ∈ N. Into each V may be refined a disjoint clopen uniform covering K with supW ∈K diam(W ) ≤ bp− j , where j ∈ N. That is, V has the uniform strict shrinking. We can consider the sequence of such shrinkings: V m+1 ⊂ V m with bm = bp−m , where m ∈ N. With each V m is associated pk(m) -nerve Nk(m) . Let k(m) ≥ −m, k(m + 1) ≤ k(m) for each m ∈ N and limm→∞ k(m) = −∞. If x is an isolated point in X, then there exists n ∈ N with max(bn , pk(n) bn ) < infy∈X\{x} ρ(x, y). Then the simplex s ⊂ Nk(m) with x ∈ s and m ≥ n is zero-dimensional over L, that is, s = {x}. By the construction of Nk for each simplex sm+1 ⊂ Nk(m+1) there exists sm ⊂ Nk(m) j with fmm+1 (sm+1 ) ⊂ sm , where fi are the bonding mappings of the inverse sequence S = {Nk(m) , fim , n). Each fmm+1 is non-stretching, since decreases the distance at least into p times and bm /bm+1 ≥ p. If x 6= y, then there exists n with max(bn , bn pk(n) ) < ρ(x, y), consequently, for each m > n there are disjoint simplexes s and s′ ⊂ Nk(m) with x ∈ s and y ∈ s′ . Therefore, there exists the uniformly continuous mapping g : X → inv − lim S, where g(x) = inv − limm {sm , fim } and sm ∋ x for each m ∈ N. Therefore, the uniformly continuous projectors fm : X → Nk(m) are defined, since for each b > 0 there exists r ∈ N such that bm pk(m+r)−k(m) < b and fm (W ) = fmm+r ◦ fm+r (W ) and diam( fm (W )) < b, where W ∈ V m+r , fm+r (W ) belongs to clopen star of the corresponding vertex v ∈ Nk(m+r) . Further as in

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Lemma IV.33 in [Isb64] we can verify that g(X) = inv − lim S and g is the uniform isomorphism. Evidently, dimL Nk(m) may be from 0 for k(m) = −m up to card(A) with card(A) = w(X). For k(m) > −m in the inverse sequence S the mappings fmm+1 may map simplexes s from Nk(m+1) into simplexes q from Nk(m) of lower dimension over L, for example, when Wm+1 ⊂ Wm , Wm ∈ V m , Wm = B(c0 (L, A j ), x, r), Wm+1 = B(c0 (L, An ), x′ , r/p), card(An ) > card(X/Rm ) ≥ dimL Nk(m) ≥ card(A j ), since dimL Nk(m+1) ≥ card(An ) for k(m+1) > −m− 1. For the complete ultra-uniform space (X, P) we can consider the base of uniform coverings {Vρn : n ∈ N, ρ ∈ P), where each Vρn is given relative to the considered ρ, ρ,k(m)

S = {Nρ,k(m) , fρ′ ,k(m′ ) , P × N}. To each Vρm there corresponds Nρ,k(m) ; ρ′ ≤ ρ if and only if ρ′ (x, y) ≤ ρ(x, y) for each x and y ∈ X; (ρ′ , m′ ) ≤ (ρ, m) if and only if ρ′ ≤ ρ and m′ ≤ m.

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Index absolute continuity of a measure §I.2.36, §II.2.31; Bochner-Kolmogorov theorems (non-Archimedean analogs) §I.2.27, §II.2.21; Borel σ-field §I.2.1; character with values in T ⊂ C §I.2.6; character with values in Ts §II.2.5; characteristic functional §I.2.6, II.2.5; completion of an algebra of subsets by a measure §I.2.1, §II.2.1; convolution of measures §I.2.11, §II.2.8; covering ring §II.2.1.1; Dirac measure §I.2.8; equivalence of measures §I.2.36, §II.2.31; Kakutani type theorems §I.3.3.1, §II.3.5; Kolmogorov theorems (non-Archimedean analogs) §§II.2.37, 39; Minlos-Sazonov type theorems §I.2.35, §II.2.30; orthogonality of measures §I.2.36, §II.2.31; positive definite function §I.2.6; projection of a measure §I.2.2, §II.2.2; pseudo-differentiable measure on a Banach space §I.4.1, §II.4.1; quasi-invariant measure . . . on a Banach space §I.3.14, §II.3.12; . . . on a topological group §III.1, §IV.1; sequence of weak distributions §I.2.2, §II.2.2; space of norm-bounded measures §I.2.1; space of Radon norm-bounded measures §I.2.1; space of µ-integrable Ks -valued functions §II.2.4. step function §II.2.1.2; tight measure §II.2.1.