Measure and Integration Theory on Infinite Dimensional Spaces

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MEASURE AND INTEGRATION THEORY ON INFINITE-DIMENSIONAL SPACES

This is Volume 48 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume

Measure and Integration Theory on Infinite-D imensional Spaces ABSTRACT HARMONIC ANALYSIS

XIA DAO-XING FUDAN UNIVERSITY SHANGHAI

Translated by Elmer 1. Brody DEPARTMENT OF MATHEMATICS THE CHINESE UNIVERSITY OF HONG KONG

@

ACADEMIC PRESS

NewYork and London

1972

COPYRIGHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR A N Y OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue,

New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD.

24/28 Oval Road, L o n d o i N W l

LIBRARY OF CONGRESS CATALOG CARD

NUMBER: 73-182639

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

FOREWORD PREFACE

Chapter I Some Supplementary Background in Measure Theory

vii ix

1

2 24 30

$1.1. Some Measure-Theoretic Concepts $1.2. Localizable Measure Spaces $1.3. The Kolmogorov Theorem $1.4. Kakutani Distance

41

Chapter II Representation of Positive Functionals and Operator Rings

49

92.1. $2.2. $2.3. $2.4.

50 62 71 79

Topological Algebras with Involution: Fundamental Concepts Representation of Positive Functionals on Seminormed Algebras Weakly Closed Operator Algebras: Fundamental Concepts Representation of Commutative Weakly Closed Operator Rings

V

vi

CONTENTS

Chapter Ill Harmonic Analysis on Groups with Quasi-Invariant Measures

103

$3.1. Basic Properties of Quasi-Invariant Measures $3.2. Characters and Quasi-Characters $3.3. Integral Representation of Positive Definite Functions on Groups $3.4. La-Fourier Transforms

105 135 164 185

Quasi-Invariant Measures and Harmonic Analysis on Linear Topological Spaces

213

Quasi-Invariant Measures on Linear Topological Spaces Linear and Quasi-Linear Functionals on Linear Spaces Continuous Positive Definite Functions on Linear Topological Spaces

214 232 254

Chapter IV $4.1. $4.2. $4.3.

Chapter V

Gaussian Measures

$5.1. Some Properties of Gaussian Measures $5.2. Equivalence and Perpendicularity of Gaussian Measures 55.3. Gaussian Measures on Linear Spaces $5.4. Fourier-Gauss Transforms

Chapter VI

Representation of Commutation Relations in Bose-Einstein Fields

280 28 1 295 31 1 326

335

Representations of the Commutation Relations in Quantum Mechanics Quasi-Invariant Measures Applied to Representations of the Commutation Relations in Bose-Einstein Fields $6.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems

369

Appendix I Background Material on Topological Groups and Linear Topological Spaces

381

$1.1. Pseudometrics, Convex Functions, and Pseudonorms $1.2. Some Properties of Semicontinuous Functions $1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces

381 384 387

Appendix II Background Material on Functional Analysis in Hilbert Spaces

394

$6.1. $6.2.

336 352

$11.1.

Operators of Hilbert-Schmidt Type, Nuclear Operators, and Equivalence Operators $11.2. Tensor Products of Hilbert Spaces $11.3. Unitary Representations of Groups

394 404 409

Notes and References to the Literature

41 3

Bibliography

417

INDEX

421

FOREWORD

T h e present book is a compendium of results which are mostly of fairly recent vintage, and the theory discussed herein is very much in a state of flux. Moreover, the original book seems to have been compiled and published rather hurriedly. Thus, there were a great number of inaccuracies in the original, ranging from typographical errors to very substantial gaps in the mathematical reasoning. I have made some effort to correct these inaccuracies; in many cases, I have altered and expanded proofs without burdening the reader with a tedious explanation of how and where the revised version deviates from the origina1. I n many cases where doubts or difficulties still remain, I have called attention to these by footnotes. However, I cannot claim either completeness or consistency in this editorial work. Especially as regards Chapters I11 and IV, I feel dubious as to how much expenditure of effort would be justified in revising or developing the theory, at least until such time as more applications may be demonstrated; in this connection, the appearance of a subsequent volume, as indicated in the author’s preface, would be most enlightening.

ELMERJ. BRODY vii

This Page Intentionally Left Blank

PREFACE

T h e study of measures and integrals on infinite-dimensional spaces arose from the theory of stochastic processes, particularly the theory of Wiener processes. I n recent years, the subject has been intimately connected with research on characteristic functionals, limit theorems, sample spaces, and generalized stochastic processes. Even more noteworthy is the fact that questions of integration on infinite-dimensional spaces have, during the last ten-odd years, appeared in many scientific fields, such as quantum mechanics, quantum field theory, statisticaI physics, thermodynamics of irreversible processes, turbulence theory, atomic reactor computations, and coding problems. However, the application of integration on infinite-dimensional spaces to these fields has encountered many profound difficulties, and a lack of adequate techniques. Thus, it seems that further study of this new subject is amply justified. Heretofore, there have appeared no introductory books on this topic, either in this country or abroad. As far as the author knows, there has been only a volume of lecture notes, “Integration of Functionals,” written by K. 0. Friedrichs, H. N. Shapiro,et al., 1957, and still unpublished. Moreover, except for Wiener integrals, the mathematical theory of measures and integrals on infinite-dimensional spaces largely began to develop only after 1956. As the mathematical background involved in the literature of this theory is rather extensive, the novice is likely to find the going somewhat difficult. Therefore, the author

X

PREFACE

has been so bold as to write the present book with the hope of smoothing the way for Chinese comrades undertaking research in this direction. This volume is primarily devoted to introducing abstract harmonic analysis. It essentially consists of three parts. T h e first part is concerned with the representation of positive functionals and operator rings (Chapter 11), which constitutes the basis of abstract harmonic analysis. Although this topic cannot be regarded as lying entirely within the domain of infinite-dimensional measure and integration theory, the two are intimately related. The second part deals with abstract harmonic analysis on pseudo-invariant measure spaces (Chapters I11 and IV) ; except for just a few theorems, the results given here were, for the most part, obtained in China. This kind of harmonic analysis may provide tools for the further investigation of measure and integration on infinite-dimensional spaces. In the third part, we discuss a mathematical problem arising in quantum field theory, i.e., the representation of commutation relations in Bose-Einstein fields (Chapter VI) ; here, applications of the theory developed in the first two parts are given. In addition, one chapter (Chapter V) is devoted to another important example of measure theory on infinite-dimensional spaces, i.e., Gaussian measures. In a subsequent volume, we shall deal with the so-called continual integral problems which appear frequently in the applications of integration theory on infinite-dimensional spaces, as well as functional variational equations and various other applications. We assume that the reader is familiar with the treatise of Halmos [I], or its equivalent, and has the basic knowledge of functional analysis which may be found in ordinary textbooks on that subject. It is also expected that the reader has some acquaintance with the basic notions of topological spaces, topological groups, and linear topological spaces ; in this connection, he may consult, for example, Guan Zhao-zhi [I]. Chapter I and Appendices I, I1 of the present book also provide some supplementary background material. Owing to the author’s limitations, and the rather short time taken to write this book, its shortcomings are undoubtedly numerous, and errors inevitable. The reader’s criticisms will be welcomed. Part of the manuscript of this book was read by Professor Zheng Ceng-tong of Zhongshan University, who offered valuable comments. T h e teachers and research students of the Functional Analysis Group, Function Theory Teaching and Research Section, Fudan University Mathematics Department, also offered valuable opinions, especially Comrade Yan Shao-zong. For these contributions, I hereby express my thanks. XIA DAO-XING

CHAPTER

SOME SUPPLEMENTARY BACKGROUND

IN MEASURE THEORY

The measure-theoretic concepts and results used in this book may, for the most part, be found in Halmos’ Measure Theory, and will be directly applied in the sequel without additional explanation. However, certain supplementary measure-theoretic results, not included in Halmos’ book, will be introduced in the present chapter; these results will also be essential in the subsequent chapters. At some points in this book, we shall require the discussion of measures which are not a-finite.1 However, non-a-finite measures in general are not well behaved (e.g., the Radon-Nikodyn theorem is not generally valid for such measures). Therefore, we shall in $1.2 investigate localizable measures, which are not necessarily a-finite, but which do retain certain desirable properties of a-finite measures. The measures ordinarTranslator’s note: The term o-finite, as used by the author, means totally o-finite in the sense of Halrnos [l]. This distinction is an important one in certain parts of this book. For example, according to the author’s terminology, a Haar measure is pseudou-tinite, but not necessarily a-finite. 1

2

I.

SUPPLEMENTARY BACKGROUND IN MEASURE THEORY

ily used on groups are localizable, so that localizable measures, in fact, constitute a fairly broad class. Some rather deeper properties of localizable measures will be introduced in $2.4. In $1.3 we shall introduce the Kolmogorov theorem. This is a fundamental theorem concerning the construction of measures on infinite-dimensional spaces from given measures on finite-dimensional spaces. We shall present this theorem in a very general form, related to the notion of a projective limit of locally convex linear topological spaces; in this form, it can be used for the construction of measures on locally convex linear topological spaces, starting from given measures on Banach spaces. In $1.4, we introduce Kakutani inner measure, which plays an important role in the study of equivalence of measures on product spaces, as well as in the study of quasi-invariant measures.

.

51.l Some Measure-Theoretic Concepts

l o Extension and Restriction of Measures We shall introduce certain generalizations of the usual notion (see Halmos [I]) of “measurable set.”

Definition l,l.l.2 Let ( G , 23) be a measurable space. Let A C G, and suppose that, for every B E 8, we have A n B E 8 . We then say that A is measurable with respect to (G, 8).We denote the totality of such measurable sets by 8. Clearly, 8 C 8,and b is a a-algebra on G. If 8 is an algebra, then 8 = B. Let f be a real (complex) function on G. If, for every Bore1 set A of the real line (complex plane), we have {g If (g)E A} E @, we say that f is a measurable function on ( G , 23). Definition 1.1.2. Let (G, S,p) be a measure space. Define a set function p on (G, 8), as follows. For A E %, @(A)= S

U p~( A

n B).

BE

We call j i the extension of p. It is easily seen that, if A E 8,then @(A)= p ( A ) . Consequently, we shall, in the sequel, denote @ simply by p, without danger of confusion. Translotor’s note: It should be recalled that, in Halmos’ terminology, a o-ring need not be a o-algebra, that is, B need not contain G itself.

8

1.1. Some Measure- Theoretic Concepts

3

In what follows, any measure space (G, 8,p) will, whenever necessary, be extended to (G, %,-p). Again, extend (G, ‘23, p) to a complete measure space (G, 8*,p*). Iff is measurable with respect to (G, 8*), we say that f is a measurable function on (G, 8 , p). If B E B and p ( B ) = 0, we call B a p-null set, or simply a null set.

Definition 1.1.3. Let (G, 8, p) be a measure space, A C G, and let 8, = { E n A I E E % } . We call 8, the restriction of 8 to A. If there exists a C E 8 such that the inner measure

we may define a set function pA on 8, as follows. For E E 8, nE)

= p(E n C).

(1.1.2)

We call pA the restriction3 of p to A.

Lemma l.l.l+4 Let (G, 8,p ) be a measure space, and let A be a subset of G satisfying condition (1.1.1) for a given C E 8. Then the restriction pa of p to A is well defined, and (A, 8,, p,) is a measure space. PROOF. We need only prove that pa is well defined; the rest is obvious. Let E, F E 8,with A n E = A nF. T o justify the definition of pA , we need only show that p ( E n C ) = ,u(F n C ) .

( 1.I .3)

We may assume that E C F, for otherwise, we could replace F by E U F. Then, from A n E = A n F and E C F, it follows that A n (F - E )

= 0,

whence C - A 3 (C n F ) - (C n E). But p*(C - A) p ( ( F n C) - ( E n C)) = 0, SO that (1.1.3) holds. 3 Translator’s note: Notice that See Halmos [l].

depends upon the choice of C .

= 0,

therefore

4

I.

SUPPLEMENTARY BACKGROUND I N MEASURE THEORY

2O The Function Space f?kz(Q) We shall have occasion to use certain abstract functions taking values in a Hilbert space. We first introduce the following notions.

Definition 1.1.4. Let H be a Hilbert space, 52 = (G, B, p) a measure space, and f an abstract function on Q such that (i) for every g E G, f (9) E H, (ii) for every u E H, the numerical valued function (f (g), u), g E G, is a measurable function on 52, and (iii) the range of values {f (g) 1 g E G) is contained in a separable subspace of H. We say that such an f is measurable, and denote the totality of such functions by M ( H , Q). I t is easily seen that M ( H , Q) forms a linear space with respect to ordinary addition of functions and multiplication by constants. Lemma 1.1.2. Let {eA, h E A} be a complete orthonormal system in the Hilbert space H. Then, a necessary and sufficient condition for f to belong to M ( H , Q) is that there exist a sequence {A,} C A and a sequence of measurable functionsAn on 52 such that (1.1.4)

PROOF. Assume that f satisfies the above condition. Then, the values off are contained in the separable subspace spanned by {eAn,n = 1,2, ...}, and (f (g), u ) = C fAn(g)(ek, u ) is measurable. Conversely, suppose that f is measurable, let M be a separable closed linear subspace containing the range off, and let {Vk) be a complete orthonormal system in M. For each k, there is a sequence (hLk') C A, such that

Therefore, the range off is contained in the separable closed linear subspace spanned by { e A p , k, n = 1, 2, ...}. Since (f,e A y ) is a measurable function on Q, and since m

the condition of the lemma is satisfied. ]

Corollary 1,1.3. If V, f E M ( H , a), then ( f ( g ) , rp(g)) is a measurable function on 52. I n particular, 11 f (g)llzis a measurable function on 52.

I . 1. Some Measure- Theoretic Concepts

5

PROOF. By Lemma 1.1.2, there is a sequence (e,,) such that (1.1.4) holds, therefore, ( f ( g ) ,d g ) ) =

1(f(g),e*n)(v(g),eAn),

whence it follows at once that ( f ( g ) , y ( g ) ) is a measurable function. ]

Definition 1.1.5. Let H be a Hilbert space, and let Q = (G, 23, p) be a measure space. Let 2 2 ( H , be the totality of functions in M ( H , SZ) which satisfy the condition

a)

( 1.1.5)

and define an inner product on !i?2(H,SZ) as follows5:

(f,PI =

1,

(1.1.6)

( f ( g ) ,9 w 449.

We let L2(SZ)(or L2(SZ))denote the usual space of measurable quadratically integrable functions on a.

Theorem 1.1.4. Let {e, , h E A } be a complete orthonormal system in H , and let HA = { f ( g )e, I f EL,(SZ)}. Then (1.1.7)

PROOF. Let f~ !P(H, SZ). By Lemma 1.1.2, there is a sequence {An} C A such that (1.1.4) holds. Since IfAk(g)I Ilf(g)ll, it follows that hk€L,(Q), that is, h k ( g )eAkE HAk. Therefore, f E W @ HA,. This shows thatf belongs to the right-hand side of (1.1.7). ] Notice that, if f,,(*) eAkE HA,, k = 1, 2 ,..., and if

0; denoting these indices by a1 ,..., a, ,..., we have

1pa(Ga n B ) = C pai(Gai n B).

(1.1.1 8)

rn

If the right-hand side of (1.1.17) is co, one can also find indices a1 ,..., a, ,... E % such that (1.1.18) holds. However, pai(Ga, nB) = p(G,, n B), hence P(B)2

C p(Gq n B ) 2 1p a i ( G a d n B)*

(1.1.19)

"i

Combining (1.1.17), (1.1.18), and (1.1.19), we obtain (1.1.16). ]

Corollary 1.1.9. Under the conditions of Lemma 1.1.8, let B E $3; then B is a p-null set if and only if, for all a E %, pe(B n G,)= 0. Example 1.1.1. Let D = (GI 23, p) be a measure space. If there is a sequence G,, rz = 1, 2,..., of disjoint sets of 23, such that 00 G = Unel G, , then {G, , n = 1, 2,...} is a partition of SZ. In general, let L? = (G, 23, p) be a measure space, and let {G, , a E 'ill} C 23 be a family of disjoint sets such that G = UaEX G, . If % is not countable, then {G, , a E %> is not necessarily a partition of D.For example, if G is the interval [0, I], and p is the ordinary Lebesgue measure on G, then {{a}, a E [0, I]} is not a partition of [0, 11. Let D, = (G, , 23, , pa), a E '2I be a family of measure spaces, and let SZ = (G, 8,p) be the direct sum of this family. For each a E %, let fa be a given measurable function on Q, = (G, , Brn , pa).Define a function f on G as follows: if g E G, , then f ( g ) =fXg).

For any Bore1 set A and any a E %, the set { g If(g)

E

A ) n Ga = (g I f a ( g )

E

A)

is measurable, hence f is measurable. Moreover, it is easy to prove the following lemma.

Lemma 1.1.10. Let D = (G, 8,p ) be the direct sum of the family of measure spaces {D,= (G, , 8,, pa), a E a}.Extend each fa eL2(Da)to a function on G by defining its values on G - G, to be zero. Then,

12

I.


60 Measures on Groups

I n the sequel, we shall frequently make use of the following type of measure on a group.

Definition 1.1.9. Let G be a group, 8 a a-ring of subsets of G, SZ = (G, 8, p ) a measure space. Suppose there is a subgroup Go of G such that each of the left (right) cosets {G, , ~1 E 'Ql) of Go in G belongs to 8 and is a-finite. Let b,, pa be the restriction of 8,p to G,, and suppose that SZ is the direct sum of the {(G, , 8, , pa), cy E a}. We then say that SZ is pseudo-a-finite. Definition 1.1.10. Let G be a topological space, (G, 8, p) a measure space. If, for every xo E G, there is a neighborhood V of xo such that V E 8 and p( V) < 03, then (G, b,p) is said to be locally finite. For example, if G is a locally compact group, the Haar measure space on G is locally finite. If (G, 8,p) is a locally finite measure space, then, for any compact subset C of G, there is an open set 0 E 8 such that C C 0 and p( 0) < 03. In fact, for each x E C there is a V, E 8 such that p( V,) < 03. By the compactness of C, there exist x1 ,..., x, such that 0 = VZlu

**.

u V Z " 3c.

< 00.

Then 0 E b and p( 0 ) \< Zrc1p( V,y)

Lemma 1.1.11. Let G be a locally compact group, 8 the a-ring generated by the totality of compact subsets of G. If (G, 8, p) is a locally finite measure space, then it is also pseudo-a-finite. PROOF. Choose any neighborhood V of the identity e of G, such that the closure C of V is compact, and C = C-l. Let Cidenote the set

cc c, I

...

and form the subgroup

u C'. m

Go =

i=l

Since C is compact, and C2is the image, under the continuous mapping (x, y) +xy, of the compact subset C x C of the product space G x G, it follows that C2 is compact. Similarly, every Ci is compact, so that the measure of Ci is finite, and hence Go is a-finite. Let {G, , E %} be the left coset system of Go in G. Clearly, each G, is also the union of countably many compact sets, and hence is


13

o-finite. We shall now prove that (G, 23, p) is the direct sum of the measure spaces {(G, , 23, , p,), a E a},where 23, , pa denotes the restriction of 8,p to G, . Since 23 is generated by the totality of compact subsets of G, we need only prove that, for any compact set K C G, P(K) =

c

P,(K

f-l

G,).

Since p,(K n G,) = p ( K n G,), it suffices to prove that there are only countably many a such that p(K n G,) > 0. In fact, since K is compact, there exists a finite set x1 ,..., x, such that

6

( x v V ) 3K.

"=l

Let x,

E

G,,

. Since C = C-1, GayC= G,,

, therefore

KC(JG,". "A

Hence, if a # a1 ,..., a,, then K n G, = 0, so that p(K n G,)

= 0.

]

Corollary 1.1.12. The Haar measure on a locally compact topological group is pseudo-a-finite. We shall next introduce some results concerning integrals with respect to Haar measures on locally compact groups. In the following three propositions, we assume that G is a locally compact group, and that SZ = (G, 8,p) is the left invariant Haar measure space on G. (orL2(Q)). Let Lemma 1.1.13. Let a €L1(SZ), (EL~(SZ)

* m)= JG 4g1) &;%) 4 4 g d

(a

(1.1.20)

*

Then a 5 EL~(SZ)(L~(SZ)). Moreover, if G is commutative, then a 5 = ( a. PROOF. If 5 €L1(SZ), then, by the Fubini theorem,

*

*

J, (J

G

I 4g1) 5(g;'g)l

dP(Zl)) d d g )

J, (J I &l) t(g;lg)I 4 4 ) ) M g l ) = J, Ia(g1)l M g 1 ) J I E(g1)l dP(&), G hence a * f EL1(Q), and II a * 5 < II a II t =

G

1 1 1

1 1 1

Ill *

(1.1.21)

r . SUPPLEMENTARY

14

BACKGROUND IN MEASURE THEORY

If 4 E L2(9),then, by the Cauchy inequality,

Also,

f2 E

L1(9),hence, substituting t2for 4 in (1.1.21), we get

Integrating both sides of (1.1.22) with respect to g, and using (1.1.23), we obtain

I, (1 I

a

471) t(gg;'g)I W l ) )

G

4.49

e II a 1:

II 5 1;

-

Consequently, a * g E L2(Q),and I1 a * 4 112 < I1 a \I1I1 4 112 * If G is commutative, then p is also right invariant, and dp(gl) = dp(gll). Setting g' = ggi', (1.1.20) becomes

whence a

*f

=4

* a.

]

Lemma 1.1.14. Let 5 , EL2(Q).Then ~ thereexist{a,} C L l ( 9 )nL2(Ln) such that (1.1.24) lim 11 an * 5 - 6 11 lim )I an * q - q 112 = 0. n+w - n+m PROOF.Let U be a neighborhood of the identity in G such that co. Form the function

p( U )
0, choose a natural number k such that Ilk < S; then x1 ,..., xp(k) is a 8-net for C . ]

Theorem 1.1.18. Let G be a complete separable metric space, b the o-algebra generated by the totality of closed sets of G, and p a finite measure on (G, 23). Then (G, 23,p) is a regular measure space. PROOF. By virtue of Lemma 1.1.17, for every natural number n, there exists a compact set C, such that (1.I .27)

Since C, is a compact Baire set, the restriction of p to C, is regular. Hence, for any E E B , there is a compact set F , E B such that F, C E n C , and 1

P((E n Cn) - Fn)
E El). This p is a probability measure, and is called the joint (probability) distribution of Xl ,..., X , . Sometimes, the function F(xl

,...,x,)

= P({w I X J w )

< x, ,

Y =

1,2,.*., n))

is called the joint distribution function of XI ,..., X , . I n particular, when n = 1, X = X , , p (or the corresponding F ) is known as the probability distribution (or distribution function) of the random variable X . It is easily seen that, if p is the joint distribution of Xl ,..., X , , then, for any bounded Baire function f,we have

90 Absolute Continuity and Singularity Relations

among Measures In this subsection, we shall prove a theorem concerning limits of Radon-Nikodym derivatives. For this purpose, we first introduce the following concept.


19

Definition 1.1.12. L e t s = (52, b,P)beaprobabilitymeasurespace, dl,...,23, a finite sequence of o-algebras such that 23, C 23, C C 23, C 23. Let x,(w), ..., xn(w) be (real) random variables on S such that xi is measurable with respect to (52, Bi).If, for any i , 1 f i < n, and any LIES,,

j

.i(W)

W W )

d

A

J

Xi+l(W)

(1.1.29)

W W ) ,

A

then {xi,23, , i = 1,..., n} is said to be a semimartingale on S . Clearly, if {xi,23, , i = 1,..., n} is a semimartingale, then, for any real number c, and any A E 23, , we have

Consequently, (max(xi(w) - c, 0), B 6 ,i = 1,..., n} is also a semimartingale. We shall also make use of certain quantities defined as follows. Let ,$, ,..., 5, be a finite sequence of real numbers, and let Y , , Y , be a pair of real numbers with rl < Y , . Let v1 be the smallest index j such that tj rl , let v2 be the smallest index j such that f j 2 r 2 ,j > v, , let v3 be the smallest index j such that rl ,j > v2 , and so on. Continuing in this manner, we get

), A E A}. ablespaces spaces {(FA Notice A', thcn then %,* SA* BE,morcovcr, moreover,forforeach eachA EA A, EA, Notice that, that, ifif hX < A', CCBT', BA* isisaa a-ring. easily that that So 23, isisa aring; ring;however, however,23,23,need neednotnot SA* a-ring. IItt follows easily be beaaa-ring. a-ring. be aa nonnegative nonnegativeset setfunction functiononon4), 23, , such Definition , such Definition 13.2, 1.3.2. Let po be that, for each A E A , the restriction of po to bA* is a measure. Then restriction of to !B,* is a measure. Then popo that, for each h E isiscalled a cylinder measure on (r, 23,). called a cylinder on So). Clearly, additive on on BSoo, ,but but need neednot notbebecountably countably Clearly, po po is is finitely additive additive on 23,. So. additive on

(r,

pA), , AAEEAA bebea afamily familyofofmeasure measure Definition ,,2B 3, A , pA), Definition 1.3.3. 1.3.3. Let (rA spaces, A } aa consistent consistent family familyofofprojections projectionsin inthethe spaces, {Pi', Ah < A'A',, A, A, A' EE A} family spaces {(FA {(rA BA), wheneverh A < and family ofof measurable measurable spaces ,,BA), AAEEAA} .} .If,If,whenever A' A'and AAEE23, 23,,,we wc have havc (1.3.3) = -'A), ( I .3.3)

{c',

%J= (Wa1

,a*.,

%").

It is easily seen that this family of mappings {Pi'}satisfies the consistency condition (1.3.1). We call {Pi'}the family of natural projections. Let r be the product space XaeB r, . For each w E r, let w, denote the ath coordinate of w , and for each h E A, h = {a1,..., a,}, let PAbe the mapping defined by w

---f

WA

= (WO1

)...,

Wan).

PAis called the natural projection of r onto PA; in particular, when h is a singleton {a}, we denote PAby P, . Clearly, {PA}and {Pi'}satisfy the relation (1.3.2). Let 23, be the totality of Bore1 cylinders in I'. We call (F, 23,) the canoptical projective limit of the family of measurable spaces (FA, BA),A E A. In this case, (r,23) is merely the product of the family

of measurable spaces (r,, %,), a E a. For each a E 8,let pa be a probability measure on (r,, S,), and for p,, on each A E A, h = {a1,..., a,}, let pA be the product measure (r,, BA).Then the family of measure spaces ( F A 23, , , pA)is consistent. Now there exists (see Halmos [I]) a product measure p on ( F , B), that is, 8, p) = ( X a ar, 9 Xmea 23, , X,P%pa). Restricting p to 23, , we obtain a cylinder measure p, , and 23, , p,) is the projective limit of the family of measure spaces ((FA, , pA),h E A}. In this case, po is countably additive on 23,.

(r,

(r,

Regarding projective limits of consistent families of measure spaces, we shall, in general, be primarily interested in the question of whether the cylinder measure po on (r,23,) is countably additive. If p, is countably additive, then it can be extended to a measure p on the O-ring 23. For the simple case described in the above example, p, is always countably additive. I n the following, we shall investigate the countable additivity of po when the rA are topological spaces.

Definition 1.3.5, Let A be a directed set, and ( r AZA), , h EA a family of topological spaces. Let P!': r,,-+ r, , A, A' E A, h < A' be a family of continuous mappings satisfying the consistency condition

I.

34


(1.3.1). Then, (P:'} is said to be a consistent family of projections in the family of topological spaces {(FA , &), A E A}. Let I' be a set and let PA: FA, A E A, be a family of mappings such that P A ( r )= FAand, whenever A, A' E A , A < A', relation (1.3.2) holds. Define a topology in r as follows. For each A E A , let

r+

ZA*= {P,'(A) 1 A E 2,).

Form 2, = UAen &*.I6 Let 2 be the weakest topology containing 2, . T h e n (r,2) is said to be al' projective limit (with respect to the family of projections {Pt'}) of the family of topological spaces { ( F AZA), , A E A}. T h e projective limit (r,2) is said to be projectively complete if the following condition is satisfied: given any sequence {A,} C A ,

< < < An < ..*)

(1.3.7)

and any sequence of nonempty sets {En)such that En is closed and compact with respect to the topology 2:, E 1 3 E 2 3 ... 3En3.*.,

then

W

(1.3.8)

En is nonempty.

Lemma 1.3.2. Let (r,2) be a projective limit of the family of HausdorfT topological spaces {(FA, 2J, X E A}. Suppose that, for each sequence {A,) C A satisfying condition (1.3.7), and each sequence of elements (5, , n = 1, 2, ...} satisfying the condition =

tnErAn,

there exists ( E

tn

for m 2 n,

(1.3.9)

r such that, for every n, PA,^

=

5,

*

(r,2)is projectively complete. PROOF. Take any sequence (An} C A satisfying condition (1.3.7), and any descending sequence of nonempty sets {E,) such that each En is Then

The elements of 2, are called open cylinders in T. Translutor's note: It should be noted that the limit of a projective system, as defined here, is not unique: given any set r' and mapping P from I" onto r, one can obtain another projective limit (Z", 2') in the obvious way, that is, 2' = P-'(2). Moreover, the condition that the projections PA(and hence also Pi') be onto means that such a limit may fail to exist, even if all the given projections Pi' are onto. Thus, this definition is far too restrictive for topological work, but it seems to suffice for the purposes of this book. Similar remarks apply to Definition 1.3.1. l6

l7

1.3. The Kolmogorov Theorem

35

closed and compact in the topology 2;. We shall prove that m En is nonempty. and, for any n Since PAmEmis compact in (rAm, m, Pi; is continuous, it follows that P,*Em = P;;PAmEmis compact (and hence closed) in (r,%, Thus, the intersection of the descending sequence of nonempty closed compact sets {PAIErn, m = I, 2,...} in ( r ASAl) l, contains a point tl. Since t1E PAIErn, m = 1,2, ..., the set P.

Define an element ( E Xaesruas follows: the amth coordinate of 5 is = 1 , 21..., and the other coordinates of .$ are chosen arbitrarily (invoking the axiom of choice). Obviously, PAn( = 5, for all n. Therefore, by Lemma I .3.2, the canonical projective limit 2)is projectively complete. ] In 94.3 we shall encounter another example in which the condition of Lemma 1.3.2 is satisfied.

varn, m

(r,

2O

Countable Additivity of Cylinder Measures

We now present the basic theorem concerning the construction of measures on infinite-dimensional spaces, stated in its most general form.

Theorem 1.3.4, Let (r,, 2,),h ELI,be a family of topological , , p,,) be a regular probability spaces, and, for each h E A , let ( F A b, measure space. Let P,"':r,,, + r,,be a family of continuous projections, consistent with respect to the family of measures {p,,}. Let 2) be a projectively complete projective limit of the family {(F, Z,,)}, with associated projections PA:r-t FA. Suppose also that (TI8, , p,) is a pro, %, , p,,)}, with the jective limit of the family of measure spaces ((r,, same associated projections P A .Then the cylinder measure po is countably additive. PROOF. By Theorem F, $9 of Halmos [I], it suffices to prove that, for any sequence of cylinders

(r,

B, 3 B, 3

- 4 4

3 B, 3

in 8,such that lim p,,(B,) = L n-1m

the intersection

n,=, €3, m

is nonempty.

> 0,

(1.3.1 1)

1.3. The Kolmogorov Theorem Now, for each n, there is a A,

37 E

A such that B,

E

Sz . We may assume ( 1.3.12)

In fact, since A is directed, there is an index A,' such that A, < A,' and A, < A,'. Then B, E St C S$g, and we may replace A, by A,'. Proceeding in the same manner for n = 3,4, ..., we obtain a sequence of indices satisfying (1.3.12). For each n, let B, = P;,'(A,), A,E 23,; By the regularity of p,, , there exists a closed compact subset C, of FA,,with C, E b,, , such that C, C A, and L tLA,,(An - Cn) < 2"+'

*

Hence, by (1.3.4), we have (1.3.13)

it follows from (1.3.13) that pO(Bn- En)< L/2. Hence, by (1.3.11), we get L PdEn) = PdBn) - tLo(Bn - En)

>2 > 0.

Therefore, En is nonempty. Moreover, En is closed and compact with respect to 2$,and El 3 E, 3 *.. 3 En 3 ..-. (1.3.14) By virtue of the projective completeness of (r,2),it follows from m m (1.3.12) and (1.3.14) that En is nonempty. But B, 3 En , m hence En is nonempty. ] The following theorem (due to Kolmogorov) deals with the case of greatest interest.

nZEl

om=,

on=,

Corollary 1.3.5.18 Let 2f be an arbitrary indexing set, and let A be the totality of nonempty finite subsets of 2f, directed by the natural ordering. For each A = {a1,..., a,} E A , let R, be an n-dimensional euclidean space, 23, the a-algebra of all Bore1 sets in R, . Denote the points of R, by x, = {xal ,...,x.,}. For each A E A, let p, be a probability measure on (R, , S,), and suppose that the measures {p, , A E A} are consistent in the following sense: given any pair A, A' E A, with l*

Regarding the terminology used here, see Example 1.3.1.

I.

38


h = {a1, a2 ,..., a,} C {pl , p2 ,...,),31 PA’({~A‘

I {x,, >...)x&

= A‘,

E E ) ) = pA(E)-

r = X a E xR, , 23 = XaEI23, , and

Let

x = {x,

(r,

and any E E BA, we have

denote the elements of

(1.3.15)

, a E a}; then, there is a unique probability measure

r

by p on

d), satisfying the following condition: for any h = {a1,..., a,} E A, and any E E 23,, , p(ix I { x q ).*.) +%,> = ( 1.3.16 )

(The measures pA, or the distribution functions on the respective spaces RA determined by the pA, are usually known as the finite-dimensional probability distributions of p.) PROOF. Each p A , being a Bore1 measure, is regular. We form the {PA}as in Example 1.3.1. Then (1.3.15) families of projections {Pi’}, shows that the family of measures {pA, h E A } is consistent in the sense of Definition 1.3.3. By Lemma 1.3.1, there exists a cylinder measure p o on (r,Bo),satisfying condition (1.3.16), and, clearly, p o is uniquely determined by (1.3.16). Let ZA be the Euclidean topology on I‘, , and let 2 be the product topology Xmexr, on I‘. T h e n (r,2)is the canonical projective limit of {(FA, ZJ, X E A} (see Example 1.3.2). By Lemma 1.3.3, (r,2)is projectively complete. Therefore, it follows by Theorem 1.3.4 that p o is countably additive. It is easily seen that 23 is the smallest 0-algebra containing !€lo, hence p o can be extended to a probability measure p on (r,23). T h e uniqueness of p follows immediately from the uniqueness of p o , ] Corollary 1.3.5 can be stated in another useful form. (In the following, the characteristic functionla of a probability distribution on a finite dimensional Euclidean space, will be referred to simply by the term “characteristic function.”)

Corollary 1.3.5’. Let rU be an arbitrary indexing set, and let 8 denote the totality of finite ordered20 sequences f = (a1,..., a,) l8 According to the Bochner-Khinchin theorem, a function #(x) of n variables x = (xl,..., x,) is the characteristic function of some probability distribution if and only if the following three conditions are satisfied: (i) # ( x ) is continuous; (ii) #(O) = 1; (iii) # is positive definite, that is, for any sequence of points dl’, ..., xIm) and arbitrary complex numbers ,..., [, , m v(x(k)

- %I &)) [ k e t

> 0.

k,l=l 2o

This means that, for any nontrivial permutation al’,a;, ...,ad of the elements ,..., a n ,the sequences ( a l ,a2 ,..., a,) and (a;, ma’ ,...,a,,‘) are regarded as distinct.

a l ,a2

1.3. The Kolmogorov Theorem

39

Tu, i = 1,..., n; n = 1, 2,...). Let {F, , 5 E S}be a family of characteristic functions, having the following properties: (i) if 5' = (al',..., a,') is a permutation of 5 = (a1,..., an),then (aiE

FP'(t,,'

,..', f,;)

= F,(ta19.m.9

L");

( 1.3.17)

(ii) for m 2 n,

)*.-,lan 0, O,***,0) = F(al,...,an)(tal ,..-,t,J

F(ml....,am)(tal

9

(1.3-18)

For each a € ' % , let R, be a 1-dimensional Euclidean space, 23, the totality of Borel sets in R, , r = R, , 23 = XmSa B, , and denote the points of I' by x = {x, , CL E a}.Then, there is a unique probability measure p on (r,B) such that, for any 5 = (a1,..., a,) E 8, FE(trrl ,...,to,,) =

J r exp i(f,pel+ ... + t,,,x,,,) 44).

(1.3.19)

PROOF. Let h be any nonempty finite subset of Tu, and let the elements of h be ordered in an arbitrary sequence 5 = ( a 1 ,a2 ,..., an). Since F , is a characteristic function, there exists a Borel measure pr on R, = Rmlx Rm2x . - * x Ransuch that FE(ful,*.*)

tan) =

J

exp i(falxal RP

+ ... + t,,~,,,) 4-+(xLyl,..., x,,).

(1.3.20)

If the indices comprising 5 are permuted, it follows from condition (1.3.17) that, after the corresponding permutation of the coordinate axes, we obtain the same measure. Moreover, (1.3.15) can be deduced from (1.3.18). In fact, if 5' = {a1,..., a,} 3 {a1,..., a,}, form the measure = PE'{(xal

J.*.?

'a,,,)

I (x>"'>@ 'a,) .l E)'

Then F&,

,-.-,f,,,o,..., 0)

By (1.3.18), (1.3.20), (1.3.21), and the uniqueness of the measure determined by a characteristic function, it follows that PP' = PP

*

I.

40


Thus, (1.3.15) holds. Using Corollary 1.3.5, we obtain a unique measure p on 23), satisfying condition (1.3.16). Since the P A ,h = {a1,..., a,}, are measurable mappings, (1.3.19) follows immediately from (1.3.16) and (1.3.20). ]

(r,

30 Sample Measure Spaces

Definition 1.3.6. Let 6 = (9,8, P) be a probability measure space, {x,( ), a E 'u} a family of (real) random variables on 6 (one sometimes calls {x,( ), a E 'u} a stochastic process on (5). Also, suppose that {xu( ), a E 'u} constitutes a determining set of functions on (9,8, P ) . For each a E 'u, let R, be a copy of the real line, and 23, the totality of Borel sets in R, . Let F C XaGa R, , and let 23 be the restriction of XUsa 123, to r. Denote the ath coordinate of x E by x, = x,(x). Suppose that p is a probability measure on (r,23) such that, for any finite set of indices a1 ,..., a n ,and any Borel set E in n-dimensional Euclidean space Rn,we have

r

P({X I (Xml *..'> Xmn) E = P({W

El)

I (%,(W>l...l

X,,(W))

E El).

( 1.3.22)

Then, (r,23, p) is said to be a sample probability measure space associated with the stochastic process {x,(w), a E %}(w E Q) on 6,and {x,(x), a E 'u} the corresponding sample process.

Lemma 1.3.6. For any stochastic process, there exists an associated sample probability measure space. PROOF. We use the notation of Definition 1.3.6. For each nonempty finite set of indices {a1,..., a,} C 'u, define a measure on R", as follows: M a

l....,mniE)= W

W

I (Xa1(~),*.*,

XNn(W))

E

El).

(1.3.23)

I t is easily verified that the family of measures {p{,l,...,,ml ,{al,..., a,} C 'u} is consistent. Let I' = XirelIR, . By Corollary 1.3.5, there exists a probability measure p on (r,123) such that (1.3.16) holds. Combining (1.3.16) and (1.3.23), we get (1.3.22). ] We conclude this section by proving a lemma, required later, which provides a criterion for the equivalence or singularity of probability measures in terms of the equivalence or singularity of their associated sample probability measures. First, we introduce a convenient modification of Definition 1.1 .7.21 a1 Translator's note: Definition 1.3.7 was introduced by the translator in order to prove Lemma 1.3.7 (which was unprovable in its original form) and to fill a gap in the author's proof of Theorem 5.2.5.

41

1.4. Kakutani Distance

Definition 1.3.7. Let QA= (G, 23, pJ, X E A , be a family of measure spaces, and 2, a family of measurable functions on (G, 23). Let 23% denote the u-algebra determined by 3 on G. Suppose that, for every set A E 23, there exists a set B , E 23% such that PA(A

nB A ) = 0

for all X E A . We then say that 2, is a joint determining set (of functions) for the family of measure spaces QA. In particular, if 2, is a determining set of functions on (G, d),then 2, is a joint determining set for any family of measure spaces (G, 23, pA).

Lemma 1.3.7. Let (r,23,p k ) , k = 1,2, be sample probability measure spaces (associated with the same stochastic process {xm(.),01 E N}) belonging to the respective probability measure spaces Gk = (52, 8, Pk), K = 1, 2. Suppose also that {xa(.), 01 E N} constitutes a joint determining set for the measure spaces 6, , k = 1,2. Then pl p2 if and only if PI Q P, , while p1 1p2 if and only if Pl 1P, . PROOF. Consider the mapping cp from Q to r defined by 9): w --f x = {xe(w), 01 E N} E r. Clearly, cp is a measurable mapping from (Q, 8) to (r,23). From (1.3.22), it is easily proved that, for any B E 23,

0 ) and {g I (dpldr) > 0, (dvldr) = O}. However, the proof still remains valid.

45

1.4. Kakutani Distance

F,,

=E -

C,F , is a null set with respect to both p and v. By the Schwarz

inequality,

(1.4.12)

= ( p ( E )v(E))”/”.

Hence, if {Ek)E 8,there is an (1.4.12),

{Fkl} E

( d E k ) v(Ek>)’/2

8o such that Fkzc Ek , and, by

2 1( p ( F k 2 )

k

v(Fk2>)”2*

k2

Consequently,

But go C 8, therefore, the inequality in (1.4.13) can be replaced by an equality. ]

30 Kakutani Inner Products of Product Measures We first consider products of finitely many finite measures.

Lemma 1.4.3. Let r k , k = 1, 2 ,..., 1 be sets, and, for each k , let be a a-algebra of sets in r k , and let p k , v k be finite measures on ( r k , B k ) . Let (Fl*, B,*) denote the product measurable space I r k , X k Z l Bk), and let p l * , vz* denote the product measures I 1 v k , respectively. Then Xk=l p k , %Jk

I

h Z * ,

”Z*)

=

n

h

k

I

(1.4.14)

vk).

k=l

Yk be a finite measure on ( r k , B k ) , such that p k < Yk , . Form the product measure y Z * = X kI = l Yk on (I‘l*,B,*). Since

PROOF. Let vk pk

<
= (x,A*y) = 0, hence Ax E H 0M . (3) Let m be a family of closed linear subspaces of H , satisfying the following conditions. (i) Every M in m is cyclic relative to %. (ii) If M , N E m and M # N , then M 1N . Let 8 denote the totality of such families m. It follows from part (1) l 4 Translator's note: Scrutiny of the proof reveals that, in general, the localizability requirement cannot be satisfied (at least, not via the author's method of proof) unless p is defined on a certain o-algebra 8' which may be much larger than 8.However, if H is separable, then one can take 8' = 8,and in this case p is, in fact, o-finite. See footnote 16.

84

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

that 5 is nonempty; in fact, m = {HE,}is one such family. 5 is partially ordered by inclusion, that is, m < m' provided that m C m'. Clearly, every totally ordered subset 5' of 5 has a supremum mE8.

m, = 7nE

8'

Therefore, by Zorn's lemma, 5 has a maximal element mo . (4) Let H' = CMMEmo 0M . We shall prove that H' = H . Obviously, the subspace H' is %-invariant. If H' # H , then H 0H' # (0). By part (2), H 0H' is also %-invariant. Choose any element toE H 0H', to# 0; as in part (I), we obtain a cyclic subspace H f o. Since Hto IM for every M E m, , it follows that the family m, w (HEJalso belongs to 5, arid is strictly greater than m, . But this contradicts the maximality of m,. 3 2.4.5. Let { H E, 5 E 9 ) be a family of subspaces PROOFOF THEOREM of H satisfying the conditions of Lemma 2.4.6, and let P, denote the projection operator from H onto H , . Since H Eis invariant under %, it follows by Lemmas 2.3.7 and 2.4.2 that P, E %' = %. For each f E 9, let

r, = {r I P,(Y) = 1). If t # t',then H , 1H,, , that is, PEP,. = 0, hence P,(y) - P,,(y) = 0 for every y E r, so that r, n is empty. For each 5 E S, define a positive functional FEon %, as follows:

0,

FE(4 = (&,

A E a.

Now, by Theorem 2.2.6, the correspondence A + A ( y ) , y ~ r is, a symmetric isometric isomorphism from % onto C ( r ) . Thus, F, may be regarded as a positive functional on C(I'), hence, by a theorem of Riesz (see, e.g., Halmos [I], §56), there exists a unique regular finite Bore1 measure pE on (r,b) such that F,(4

=

14 4 r

dCL,(Y)*

Since P,t = t, we have FE(P,A)= F,(A) for all A F,(4

=

1 PdY) 4) r

dCLE(Y) =

E

%, hence

1 4)44Y). re

(2.4.5)

Since C ( r , )is dense inL1(r, 23, p,), it follows easily that p L r ( r T,) = 0. Let B E= { B I B E b,B C rE}. We thus obtain a family of finite measure spaces Q, = (r,, 23, , pE),5 E 9.

2.4. Representation of Commutative Weakly Closed Operator Rings

85

Consider the dense linear subspace M , = { A t I A E '%I of H , , and the space C(F,) of continuous complex-valued functions on F,. We define a mapping ye from M , onto15 C(F,), as follows: if A E 8, then

By (2.4.5), (2.4.7)

for any A , B E 9I. If we regard C(T,) as a subspace of L2(9,),then relation (2.4.7) means that y, is isometric. Since M , is dense in H , and C ( r , ) is dense in LZ(SZ,) (see Halmos [I]), it follows that ye can be uniquely extended to a unitary operator from H , onto L2(Q,). Let A , = ycAcpcl. Then

In fact, from (2.4.6) and the relation ( A B ) ( y )= A ( y ) B(y), we see at once that (2.4.8) holds when f ( y ) = B ( y ) , B E 9I. By virtue of the density of C(f,)in L2(Q,), this implies that (2.4.8) holds for all f E L2(Q,). Let

ro= r - C€B u r, ,

6,= { B I B E 6 ,B c ro},

Form the direct sum SZ = (r,6, p ) of the measure spaces16 {SZ, , 6 E E} and SZ, . By Example 1.2.1 and Theorem 1.2.6, SZ is localizable. Moreover, by Lemma 1.1.10, (2.4.9)

Since F - r, = { y 1 P&) = 0). it follows that F, is both open and closed in F. Hence, any function in C ( r , ) can be extended to a continuous function on r by simply = IC } (r), assigning the value zero to all points of r - I', . By Theorem 2.2.6, (A(y)l A E ? therefore the image of vt does cover all of C ( r , ) . l8 Tronslutor's note: There is a difficulty here. If the indexing set S is uncountable, then the o-algebra obtained by taking the direct s u m of these measure spaces will, in general, be larger than 23. If one regards each p, as defined on 9, and simply defines p = &E p, , then one obtains the desired result, except that p may not be localizable. Of course, if H is separable, then 8 is countable, and p is, in fact, o-finite.

86

11. REPRESENTATION

OF POSITIVE FUNCTIONALS AND OPERATOR RINGS

We now combine the family of mappings mapping y , as follows. If x E H, define

{vS, 5 E S} to form

a single

(2.4.10)

I t is clear from (2.4.3) and (2.4.9) that 9 is a unitary operator from H to L2(Q). When f E L2(QS), we have q r ' f = q~ry, hence, by (2.4.8),

But since the operators appearing on both sides of (2.4.1 1) are bounded, it follows that (2.4.1 1) holds for all f E L2(B). Let C = (p4v-l I A E W}; then C C %l(B).Since (2.4.10) is a unitary operator, it follows from the remarks in part 4" of $2.3 that C is a maximal commutative weakly closed operator ring over L2(sZ).Obviously, %l(Q) is symmetric and commutative, hence, by Lemma 2.4.1, the weakly closed ring R(%l(sZ))generated by 'iUl(sZ) is also commutative. Since R(%l(Q))3 9l(Q)3 C, the maximality of C implies that R('iUl(sZ))=

m(Q)= c. ]

While continuing to use the notation of Theorem 2.4.5, we now consider the case of uniform multiplicity k.

Corollary 2.4.7. Let W be a commutative weakly closed operator ring having uniform multiplicity k over H . Then, there exists a measure p on (T, b) such that B = (T,b,p ) is a localizable measure space, and there is a unitary operator q~ from H onto Qk2(Q) such that, for every

A E W, (vAv-l)f(y)

= A(y)f(y),

f E Qk2(Q),

moreover, the correspondence

is an isomorphism from Y l onto %Jlk(Q). T h e above corollary may be easily deduced from Definition 2.4.2, Theorem 2.4.5 and Theorem 1.1.4; the details are left to the reader.

Corollary 2.4.8. Let W be a commutative17 weakly closed operator ring over the Hilbert space H. If '3 has a cyclic element in H, then there l7

Notice here that we do not hypothesize the maximality of N.


87

exists a finite measure p on (I',23) and a unitary operator y from H onto L2(F,23, p ) such that ?A?-'f(Y)

=

A(Y)f(Y)

for every A E 21 and f E L 2 ( r ,23, p), moreover, the correspondence A -+ rp4v-l is an isomorphism from 'u onto 'iUl(I',23, p ) . PROOF. By assumption, H = Ht , where 5 is a cyclic element of H relative to a.Thus, in the proof of Theorem 2.4.5, P , = I , and we obtain a finite regular measure space 52 = (I',B, p ) and an isomorphism of 2l onto C(I'), regarded as a subring of the multiplication algebra 'iUl(sZ). Using the density of C(I') (qua function space) in L2(9),and the fact that C ( r ) (qua operator ring) is weakly closed in 23(L2(sZ)),it is easily verified that C(I') = 2Jl(52). 3

Corollary 2.4.9. Let 2l be a maximal commutative weakly closed operator ring over a separable Hilbert space H . Then '2I has a cyclic element in H . PROOF. Since H is separable, the indexing set E in the proof of Theorem 2.4.5 is, in this case, finite or countable. Consequently, Q is the direct sum of finitely many or countably many finite measure spaces (r,, 23, ,p,), k = 0, 1, 2,..., and hence is o-finite. Thus, (p4v-l I A E a} has a cyclic element in L2(Q),namely,

where Crk(,) denotes the characteristic function of I',. It follows immediately that '2I has a cyclic element in H . ]

Lemma 2.4.10. If 52 = (G, 8,p) is a finite measure space, then the multiplication algebra il.R(sZ) over L2(Q) is maximal commutative weakly closed. PROOF. Let u E (ml(Q))'; then ul cL2(Q).Let ul be represented by the measurable function u(g) on Q. For any bounded measurable function q on Q, we have uv

=

(uv)]

=

(v4l

=

(2.4.12)

P(d 4d9

hence JG / u(g)T(g)l2dp(g) < // u 112 I! q j 112. Given any positive number E , let q ( g ) be the characteristic function of the set {g 1 1 u(g)I2 2 11 u 112 c} = E. Applying the preceding inequality, we have pfE)(llu /I2 €1d I1 u 11' *P(E), hence p ( E ) = 0. Thus, / u(g)! < (1 u // almost everywhere. Consequently

+

+

88

11. REPRESENTATION OF

POSITIVE FUNCTIONALS AND OPERATOR RINGS

for any q~ €L2(Q),both sides of (2.4.12) belong to L2(f2). Since the totality of bounded measurable functions q~ is dense in L2(Q),it follows that uqI(g) = u(g)qI(g)for every q~ EL~(Q),that is, u E mZ(Q). ] We can now state a sufficient condition for the maximality of a commutative weakly closed operator ring.

Corollary 2.4.1 1. Let 2l be a commutative weakly closed operator is ring over a Hilbert space H . If 2l has a cyclic element in H , then QI maximal commutative weakly closed. PROOF. By Corollary 2.4.8, M is unitarily equivalent to a multiplication algebra 9X(Q), where 52 is a finite measure space. By Lemma 2.4.10, %R(Q)is maximal commutative weakly closed, therefore (11 is also maximal commutative weakly closed. ] We now consider a more general situation.

Theorem 2.4.12, Let Q = (G, 8 , p) be a localizable measure space. Then the multiplication algebra m(Q) over L2(Q)is a maximal commutative weakly closed operator ring. PROOF. Obviously,m( Q) is a symmetric operator algebra, and isclosed in the uniform topology. We need only prove that (!JJl(Q)’)p C m(Q). For then, given any self-adjoint operator A E ‘3n(sZ)’, we know by Corollary 2.3.4 that all the projections PA associated with A belong to (%R(Q)’)p, and hence to m(Q),so again, by Corollary 2.3.4, A E~X(Q). But since rol(Q) is symmetric, this implies that YX(Q)’ C %Jl(Q); obviously, since B(Q)is commutative, %(Q) C%(Q)‘. Thus, we obtain ~ ( Q= ) %R(Q)’, whence it follows by Lemma 2.4.2 that W(Q) is a maximal commutative weakly closed operator algebra. Let P E ( ~ ( Q ) ’ ) p , and consider the collection of sets

5

= {E

I E E B , p ( E ) < 03).

For each E E 5, let C E denote the characteristic function of E, and PE the multiplication operator corresponding to C, : (pEf)(g) = CE(g)f(g),

g

G,

fEL2(’)*

Denote the restrictions of 8 , p to E by B E , p E , respectively. Since p ( E ) < 03, Q, = (E, B E ,p E ) is a finite measure space. Regarding the functions in L2(sZE)as functions in L2(Q) which vanish outside of E, we have L2(QE)= PEL2(sZ),and it is clear that m(52,) is then just the restriction of %R(Q) to L2(Q,).Since P E (%R(Q)’)P, the projections P and P , commute, hence PPE is a projection operator on L2(9,), moreover, since P E ~ m ( Q E and ) PE%R(Q)‘,we have P P E ~ m ( Q E )However, ‘. by


89

Lemma 2.4.10, )132(QE)= )132(QE)',hence PPE E '9X(QE), that is, there exists a set A, E B E such that (ppEf

) ( g ) = CA,&)f

(g),

g

E,

f EL2(QE)*

Obviously, the above formula is also valid for any f €L2(Q),therefore PPE is the projection operator corresponding to AE E b. Now, G VAE% A , and so, by Lemma 1.2.5, I = V E E %PE , hence, P = V E E PPE S ; moreover, by the localizability of Q, there exists Q E 23 such that

-

Q=

V ~ E . EE5

Again, by Lemma 1.2.5, P is the projection operator P , corresponding to the set Q, hence P~)132(Q).]

Corollary 2.4.13. Let Q = (G, b,p) be a localizable measure space,

K a cardinal number. Then the multiplication algebra YlIm,(Q)over f!,z(Q) is a commutative weakly closed operator ring, and has uniform multiplicity k. This corollary is a direct consequence of Theorem 2.4.12; the details of the proof are left to the reader.

30 Some Properties of Localizable Measure Spaces We shall now make use of Theorem 2.4.12 to study in further detail the properties of localizable measure spaces.

Theorem 2.4.14. Let Q = (G, b,p) be a localizable measure space, and let F be any continuous linear functional onLl(Q). Then, there exists an essentially bounded measurable function F(g), g E G, such that

moreover, J j F 11 = 1) F Ijm. PROOF. Define a bilinear functional on L2(sZ),as follows: F(v, 4)= F(v$),

If

V?

4 EL2(Q)*

v, # E LZ(Q),then v$ E L1(Q), hence F ( y , #) is well I F(v, 411

< II F II I/ v

112

II

*

(2.4.14)

defined. Moreover,

112 >

hence F ( y , #) is continuous. Therefore, by a well known theorem, there

90

11. REPRESENTATION

OF POSITIVE FUNCTIONALS A N D OPERATOR RINGS

exists a bounded linear operator A on L2(Q)such that 11 A F(P, 1CI)

=

(A%9%

7 1

I( < 11 F I( and

ttr EWQ2)-

Let B E 'Dl(52), and suppose that B corresponds to the bounded measurable function b(g). Then, for any rp, I,4 E L ~ ( Q ) , (A%,

$1 = F ( b &

= (AT,

64) = (BA9J94 1 9

that is, A and B commute. Thus, AE%T(Q)'.By Theorem 2.4.12, %T(52) is maximal commutative weakly closed, hence A E %T(Q). Therefore, there exists an essentially bounded measurable function F ( g ) , g E G such that Arp(g) = F ( g ) q ( g ) , whence F(&)

=

1 F(g) G

(2.4.15)

d g ) $(g) d P k )

for any v, th, E L ~ ( SBut, ~ ) .for anyfELl(Q), the functions I,4 = If l /z and If I-lI2 belong to L2(Q).Substituting these functions into (2.4.15), we obtain formula (2.4.13). From (2.4.13), it follows at once that IIFII I(FII, , and since /IF/loo = /I A 11 /IF11, we get /I F /I = I1 F Ilm * 1 rp = f *

0 is arbitrary, we conclude that (2.4.29) for almost all g E G (relative to p3). An entirely similar argument shows that the reverse inequality also holds for almost all g . Hence, (2.4.25) is valid for almost all g E G (relative to the measure p3). ]

Corollary 2.4.18, Let Q, = (G, 8 , pL), k = 1,2, be equivalent localizable measure spaces. Then the Radon-Nikodym derivative dpl(g)/dpz(g)of p1 with respect to pz may be chosen so that

PROOF. Taking p3 = p1 in Theorem 2.4.17, we may choose d k ( g ) / d p 3 ( g )= hence 3

(2.4.30) g )co, it follows from (2.4.30) that for almost all g. Since d p ~ ( g ) / d p ~ (
0 almost everywhere. Thus, we need only alter the values of dp,(g)/dp,(g) on a null set and so obtain the required function. ]

Corollary 2.4.19.

Under the hypotheses of Theorem 2.4.15, the

mapping T

2

'p

-

p?(dcLlldr.L2)'/2

is an isometric operator from L2(Ql) into L2(Qz). PROOF. Let y , E L ~ ( Qthen ~);

However, y J

E

L1(Q,), hence, applying (2.4.16),we obtain

=

(TV, T#),

that is, T is isometric. 3 We shall now establish the general form of an operator in [2Jl,(f2)']u, where Q is a localizable measure space. For convenience, we shall explicitly discuss only the case k No. Let {eA, h E A } be an orthonormal basis for the k-dimensional Hilbert space H , , let HA = {yeAI y E L2(Q)},and let PAbe the projection operator from Qkz(Q) onto HA. I t is easily verified that PAE [YJl,(Q)]', moreover, by (1.1.7), we have

0, there is a %-neighborhood V of the identity in 8 (we may suppose that V = V-l) such that MI@) < (3.1.34) for all h E V. From (3.1.34) and the last inequality in the proof of Lemma 3.1.19, it follows that l(P(h-1B))112- (P(B))l/zI

0.” Since hE - E is a null set, we have p ( h E nF )

= 0.

(3.1.40)

Moreover, since v and p are equivalent, v(E) > 0. l6 Trunslutor’s note: Note the unfortunate fact that strong ergodicity, as defined here, is a weaker property than ergodicity. l7 Translutor’s note: This inference seems to require that 2 3 be a o-algebra. Alternatively, one could modify the definition of ergodicity as follows: a measure space is ergodic provided that there do not exist two disjoint quasi-invariant sets which are both of positive measure.

132

111.

GROUPS WITH QUASI-INVARIANT MEASURES

Consider the measure pI on (G, B), defined by /Ll(A) = v(A-I),

A E 8.

As usual, we denote the characteristic function of any set ID by Ca One easily calculates that

.

By Fubini's Theorem (see Halmos [l], $36, Theorem B),

(3.1.41)

But, in view of (3.1.40), the right-hand side of (3.1.41) is zero, which contradicts v(E) p ( F ) > 0. We conclude that p is ergodic. 3 Remark. As particular cases of Lemma 3.1.31, we have (i) any leftinvariant Haar measure on a locally compact topological group, and x pLnof left-invariant Haar measures p1 ,..., p, (ii) any product p1 x on locally compact topological groups G, ,..., G, , respectively. Next, we introduce a construction which will be used in the sequel.

Theorem 3.1.32. Let {G, , a E 2l} be a family of locally compact topological groups, let 8, be the o-ring generated by the totality of compact sets in G, , and let Q, = (G, , B , , p,) be a probability18 measure space which is left quasi-invariant under G, . Let G = Xaal G, ; then G is a group with respect to the multiplication operation defined as follows: (go!, a E W

h ,

9

01

w

= {g,h,

, a E 2%

Let 0 be the subgroup of G consisting of all elements g = {g, , a E 2l} such that g, differs from the identity element of G, for at most a finite number of indices a. Let 8 = XaElb,, p = Xaal pol; then, the measure space Q = (G, 8 ,p ) is ergodic with respect to the group of left translations corresponding to 0. In particular, this implies that b, is a o-algebra.

3.1. Basic Properties of Quasi-Invariant Measures

133

PROOF. Let h = {a1 ,..., a,} be an arbitrary finite subset of CU. Form the Cartesian product G,

=

G,, x G,, x ... x Gmm,

and define multiplication in GA by the rule {g&1

)'''9

gmn>(h,l

>.'.I

=

{gcYlhml

>-"f

gcXfihmn}.

If GA is given the product topology, then it is easily shown that GA is also a locally compact topological group. Let 23, = Bml x 8,,x x Ban, PA = pa, x " * x pan; then, using Fubini's theorem, it is easily proved that SZ, = (G, , 23, ,p,) is a probability measure space which is left quasi-invariant under G, . By Corollary 3.1.6, each p, is equivalent to a left-invariant Haar measure v, on G, . Hence, p,, is equivalent to the measure vU1x - - - x vNnon 23, . By the remark following Lemma 3.1.3 1, vU1x x van is ergodic with respect to the left translations of G A , hence the same is true of pA. If CU is finite, then G, = 6 ,hence, it only a * .

remains to consider the case where 'u is infinite. Suppose that E is any set in 23 which is quasi-invariant under 6. Now, there is a sequence {a,} C CU, such that E is the direct product of m )(olg(,n) G, and some set in b a n .Consequently, we may as well suppose that CU itself is countable, for example, CU = (1, 2,..., n ,...}. Let A, = (1, 2,..., n}. If f(gA,> EL2(QAn),

then we may regardf(.) as a function in L2(Q),that is, g +f(gAn), g E G, {gAn = (g, ,g, ,..., g,)}. Thus, L2(QAn) is imbedded as a closed linear subspace of L2(Q);let P n )denote the corresponding projection operator. If S E L2(!2),X E L2(QAn), then, since P(,) is self-adjoint, we have

p'"'SkAn)

xx) 1 dpA,(ghn) =

C

p'n'S(gAn)x(g,n) 'pk)

I n particular, when the function S is nonnegative, formula (3.1.42) holds for any nonnegative measurable function X on Q A - ,provided that (Pcn)S)X ~ L l ( i 2 , ~I n) .fact, for such an X , we have X,

= min(X,

m)€L2(QAn),

134

111.


hence, setting X = X , in (3.1.42), and letting m + 03, we obtain the desired result. Now, let S be the characteristic function of the set E; obviously, S E L2(Q). Write S(,) = P n ) ( S ) . Choose an arbitrary element h = {h, ,..., h,} E GAn,let Y be any nonnegative measurable function on QA such that Scn)(hgAn) Y(gAn)€L1(QAn), and define

we have Scn)X E L1(QAfl). Hence, using (3.1.42) and (3.1.43), we get S'n)(hgAn)'@An)

dpA,(gAn)

Let k = {ha , a E m} be the element of 8 whose coordinates are k, = h, for a = I, ...,n and k, = e, (the unit element of G,) for 01 > n. Then, by virtue of (1.1,13), the above relation may be written as

(3.1.44)

-

Since E is a quasi-invariant set, the functions S(g) and S(kg)are equal almost everywhere. Therefore, applying the change of variable g hg in (3.1.44), we obtain S(n)(hgAn)'(gAn) dpAn(gA,) =

I,

=

I,

S(g) y(gAn) dr*.(g) S'"'(gAn)

dpAflkA,),

3.2. Characters and Quasi-Characters

135

whence it follows easily that Scn)(hgAn) and S(")(g,,,)are equal almost everywhere. Since h is an arbitrary element of G A n ,and since SZAn is ergodic with respect to the left translations of G,,, , it is not difficult to deduce that S(")(gAn) is almost everywhere equal to some constant M, . But according to Lemma 1.1.7, the sequence of projections {Pn)} converges strongly to I, hence S

=

lim P(n)S = lim M,, , n+m

n+m

which is obviously also equal to some constant almost everywhere. This means that C, is either equal to zero almost everywhere or equal to 1 almost everywhere, that is, p ( E ) = 0 or p(G - E) = 0. Thus, SZ is ergodic with respect to the group of left translations 8 . ]


lo Definition and Basic Properties of Characters The class of functions known as characters is of basic importance for harmonic analysis on groups. Their definition is as follows.

Definition 3.2.1. A complex-valued function a on a group G is said to be a character, provided (i) I a(g)l = 1 for all g E G, and (ii) a(gh) = a ( g ) a(h) for all g , h E G. If we let e denote the unit element of G, then any character a of G satisfies the condition (iii) a(e) = 1. I n fact, by (ii), a(e)2 = a(e2) = a(e), and by (i), a(e) # 0, hence a(e) = 1. In the present section, we shall use C to denote the multiplicative group of complex numbers of unit modulus; thus, a character a of a group G is just a homomorphism of G into C: a :

g+a(g),

gEG.

Let a and /3 be any two characters of the group G. We define the product a/3 by the ordinary rule for multiplication of functions:

Obviously, a/3 is also a character of G. The totality of characters of G clearly forms a commutative group with respect to multiplication. This group is known as the character group of G (or the algebraic dual of G ) ; we denote it by G'. The unit element of G' is the constant function 1.

136

111. GROUPS

WITH QUASI-INVARIANT MEASURES

Example 3.2.1. Let I be the additive group of all integers. For each c in C, define a character /Ic of I as follows: /3&)

= cn,

n E I.

The correspondence c -+ /Ic is obviously an injective homomorphism from C into I'. Moreover, given any a E 1', we have a = / I c , where c = a( I). Thus, the correspondence c -+ /Ic is an isomorphism from C onto 1'. Using this isomorphism, we may also consider C and I' as identical: I' = C.

Theorem 3.2.1. (Character Extension Theorem). Let G be a commutative group, GI a subgroup of G, and a, a character of G I , Then, there exists a character a of G such that a(g) = al(g) for all g E Gl. PROOF. If G # G,, choose any g, E G - G,, and let G, be the smallest subgroup of G containing g, and G, . We distinguish two cases. (i) For every natural number n, gln 2 G, . I n this case, every element of G, is uniquely expressible in the form ggln,

gE Gl 1 n

= 0,

&I, Z t L .

*

Choose any c1 E C, and define a character a , on G, as follows: 4gg1")

=

al(g)Cln-

(3.2.1)

I t is easily seen that a2 is an extension of a , . (ii) There is a natural number n such that glnE G I . Let n, be the smallest such number. Then, every element of G, is uniquely expressible in the form gglR,

gEG1, n = 0 , 1 , 2 ,..., n1-1.

Choose a complex number c1 such that cy1 = a,(g?), and define a character a, on G, as in formula (3.2.1). Again, it is easily verified that a, is an extension of a, . Thus, in either of the two possible cases, a, can be extended to G,. Using Zorn's lemma, we can extend a, to a character a on G. ]

Example 3.2.2. Let 9 be the additive group of all rational numbers, let t be an arbitrary real number, and consider the character of B? defined by g E B. q(g) = e i t g , The multiplicative group of all characters of this form is obviously isomorphic with the additive group of real numbers R. However, this

3.2. Characters and Quasi- Characters

137

group is not the whole of 9'. I n fact, choose any sequence of natural numbers (9,) such that qn divides qn+l, 29, < qn+l, and such that every natural number divides at least one of the q , . Clearly, such a sequence does exist. Define another sequence of natural numbers {p,} by the recursion formula

po Obviously, p , integer k,

-+

= 0;

p,

= pn-l

+ 2qn_,,

co.Now, form a character

n 01

=

f , 2,... .

on 9 as follows: for each

We must first show that 01 is well defined. If k/q, = k'/q,# n < n', then p,. - p, is an integral multiple of qn , hence

&k'

=&k

4n'

4n

=&

, and, say,

k + r,

4n

where r is an integer. Thus, 01 is well defined, and now one may easily verify that 01 is a character. Now, we assert that this 01 cannot be equal to any a t . For suppose, on the contrary, that a: = tyf for some t. Then, since a(l/qn)= a f ( l / q n ) it , follows that

is an integer. However, when n is sufficiently large, we have I t / 2 I~< i q , , and hence I p, - (t/277)1/qn< 1 , which is possible only if p, - (t/277)= 0. But this cannot hold for two different values of n. We conclude that 01 # cxf for any t, and thus 9'# R.

Example 3.2.3. Let R be the additive group of all real numbers. For each fixed real number t, we have the character at(.)

=

eitz,

x E R.

Denote the totality of such characters by R*. We assert that R" # R'. I n fact, in Example 3.2.2 we constructed a character 01 on the subgroup 9 of R,such that 01 is not of the form a ffor any t E R; using the character extension theorem, we may extend this 01 to a character of R, and of course this extension is also different from every a t . Actually, one can even extend a character of the form a t on 9 to a character on R which is not of the form a t . For example, choose any irrational number xo ,

111.

138


and let G1be the subgroup of R consisting of all numbers of the form mxo+x,

Let /3 be the character of

mEI, X E W .

G1defined by B(m0

+ x)

=

(-1)".

Then, any extension of jl to a character of R is obviously not of the form a f . In a similar manner, one can prove that C' # I , that is, there exist characters of C which are not of the form ol,(a)

= an,

n €1.

When G is a topological group, we shall be primarily interested in continuous characters of G, that is, characters which are continuous functions on G. T h e totality of continuous characters of G is called the dual of GI and will be denoted by G*. Obviously, G* is a subgroup of G'. Of course, if G is a discrete topological group, then every character of G is continuous, so that G* = G'. However, for an arbitrary topological group GI in general, G* and G' do not coincide.

Lemma 3.2.2. Let /3 be a character of the topological group G. If jl is continuous at the identity e of GI then /3 E G*. PROOF. Let ho be any element of G, and let E be an arbitrary positive number. By hypothesis, there exists a neighborhood V of e such that I jl(g) - /3(e)l < t for all g E V . Hence, if h belongs to the neighborhood Vho of h a , then

I B(h) - P(h0)l = I P ( h W - B(e)l < E. Thus, ha is also a continuity point of /3. J Let 23 be a o-algebra consisting of certain subsets of a group GI and let G' denote the totality of 23-measurable characters of G. Obviously, G' is also a subgroup of G'. If 23 consists of all the subsets of G, then of course G' = G'. On the other hand, if 23 contains only the null set and G, then G5 consists of just the single character 1. If G is a topological group and if 23 contains all the closed subsets of G, then every continuous character of G is %-measurable, that is, G* C G'. I n the following example, we actually have G* = GB.

Example 3.2.4. Let R be the additive group of real numbers with the Euclidean topology. Then, every continuous character on the topological group R is of the form m t ( x ) = eifz, x E R. Moreover, if 23


139

is the 0-algebra generated by the totality of closed subsets of R, then

RB = R*. In fact, let a E RB,and consider the Lebesgue integral

1

m

X

=

a(x)e-”dx.

0

Since

1

m

a(h)X =

a(h

+ x)e+

m

dx

=

eh

a(x)ecz dx,

(3.2.2)

h

0

it follows that a(h) is continuous. Thus, RB = R*. Furthermore, using (3.2.2), we obtain

hence, as h -+ 0, a(h) - 1 +I-- 1 h A’

therefore a’(0) exists. Moreover, since

a’(g) exists and is equal to a‘(0) a(g). Integrating, we obtain a(g) = e a ’ ( O ) g .

Therefore, since I a(g)l = 1, it follows that a’(0) is a pure imaginary number, which we call it, t E R. Thus, every measurable character of R is of the form a t . The preceding example may also be regarded as a special case of Lemma 11.3.2 of Appendix 11.

2* Quasi-characters In what follows, we shall require a class of functions more general than measurable characters, namely, quasi-characters. We shall not describe the concept of a quasi-character in its most complete generality; for our purposes, it suffices to restrict our considerations to groups.

Definition 3.2.2. Let (G, B, p ) be a measure space which is quasiinvariant under a group of measurable transformations 8 . Let a be a

140

111.


measurable function on (G, ‘B), satisfying the following conditions. (i) I a(g)l = 1 for all g E G; (ii) for every fixed h E 6, the function gEG

4g)/a(g),

is equal to a certain constant &(h) almost everywhere (relative to the measure p). Then, we say that a is a puasi-chara~ter~~ of (G, 8,p) relative to 6 . First, we observe that the function B(h), h E 6, is a character of 0. In fact, we obviously have I 8(h)l = 1, moreover, for any h, x E 6,

hence, it follows from the quasi-invariance of p that 8(hx) = 8(h)~ ( X ) . ~ O We call B the character of 6 induced by the quasi-character a; the totality of such induced characters will be denoted by 6”.We also refer to 01 as a quasi-character corresponding to 8. We shall regard two quasi-characters which are equal almost everywhere (relative to p ) as identical. If two quasi-characters differ only by a constant factor (i.e., if the ratio of these two functions is equal almost everywhere to some constant), then we say that they are similar. We proceed to state some simple properties of quasi-characters. I. Similar quasi-characters induce the same character. 11. If a is a quasi-character and c is a complex number with I c I = 1, then ca is also a quasi-character. 111. If the measure p is ergodic with respect to 6, then quasicharacters which correspond to the same character are similar. In fact, suppose that the quasi-characters a1 and a2 correspond to the same character of 6, and consider the function a = ala;l. Then, for any h E 6, we have a(hg) = a(g) for almost all g E G. Consequently, l o A quasi-character can be regarded as an eigenfunction of a certain operator. In fact, consider the totality U ( G ) of %-measurable complex-valued functions 01 on G such that I a(g)l = 1 for all g ; functions in U ( G )which are equal almost everywhere ( p ) are to be regarded as identical. For each element h of 6, define an operator Thon U(G)as follows:

ThO1(g) = a(h-’g),

01

E

U ( G ) , h E 6.

Then, the correspondence h + T his a representation of 6 in U(G)[i.e., a homomorphism of 6 into a group of operators on U(G)].A quasi-character 01 is just a common eigenfunction of all the operators T h, h E 6, and a(h) is the eigenvalue of the operator Th-1, corresponding to the eigenfunction 01. Translator’s note: To draw this conclusion, one must assume that the measure p is not identically zero.


141

if y is any arc on the unit circle of the complex plane, then { g I a(g) E y } is a quasi-invariant set. But if a(g) were not equal to some constant almost everywhere, then there would exist two disjoint circular arcs y1 ,yz such that both of the (disjoint) sets ( g I a(g) E yl>and ( g I a(g) E y2) are of positive measure. This contradicts the ergodicity of p. IV. If all the quasi characters corresponding to any given character of 6 are similar, then p is ergodic with respect to 6. I n fact, if p is not ergodic, then let A be a quasi-invariant set in G such that p ( A ) > 0 and p(G - A) > 0.21 Define a function a: as follows: a:(g) = -1

for g E A ,

.(g) = 1

for g E G - A .

I t is easily seen that a: is a quasi-character which induces the character 1 on 6. But 01 is not similar to the quasi-character 1 of G. We shall denote the totality of quasi-characters by Gu. Clearly, Gu forms a group with respect to ordinary multiplication of functions. T h e totality of quasi-characters which correspond to the character 1 of 6 will be denoted by '$I. It is easily verified that % is a subgroup of G.. Moreover, it follows from properties I-IV that '$I is the tbtality of constant functions c, c E C, if and only if p is ergodic with respect to G. Consider the factor group Go&= G'/%. For each 77 E GOP, choose a representative element a in 7.T h e mapping

s:

(3.2.3)

q+&

is clearly an isomorphism from GOuonto

6~.

Example 3.2.5. Let G be a group, let 6 be the totality of left translations of G, let (G, b,p ) be a finite measure space which is quasiinvariant under @, and suppose that the correspondence (g,h) -+gh is a measurable mapping from (G x G, 23 x 23) to (G, 23). Then, the group of quasi-characters Gu is just the totality of functions of the form c&,

C E C , & E GB.

PROOF. For any a: E GP,the functions a(hg) and a:(g), (g, h) E G x G, are both measurable with respect to 23 x 8 , hence, the same is true *l

Translator's note: Again, see footnote 17 concerning the proof of Lemma 3.1.31.

142

111.


of their quotient ai(hg)/cx(g). Since p is finite, we may assume thatz2 p(G) = 1. By Fubini’s theorem, we know that

is a measurable function on (G, b), that is, 2 E G’. I n particular, Z(h), regarded as a function of the two variables ( g , h), is also measurable, hence

is a measurable set. Since

for every h E G, it follows from Fubini’s theorem that ( p x p ) ( E ) = 0, and therefore

for almost every fixed value of g . We arbitrarily choose an element g = go such that the above equation holds. T h e n

4ko)

=

.(go)

for almost all h in G. Writing hg, = h, , it follows by the quasi-invariance of p that

for almost every h, , that is,

a(go)/a(go). ] Example 3.2.6. Let G be a locally compact group, Q the group of all left-translations of G, and (G, b,v) a left-invariant Haar measure space. Then, given any quasi-character a of (G, 8,v) relative to 6, there exists a complex constant c E C such that a ( g ) = cZ(g) for almost every g E G. PROOF. Let Go be a subgroup of G constructed as in the proof of Lemma 1.1.1 1, and let Bo , vo denote the restrictions to Go of b, v, respectively. Then, the measure space (Go, S o ,y o ) is a-finite, hence, aa

zero.

ai

= cZ,

where

c =

Translator’s note: Again, it is implicitly assumed that the measure p is not identically


I43

there exists a finite measure p on ( G o ,So),such that p is equivalent to y o . Now, let a be any quasi-character of (G, 8,Y ) , relative to 8. Then, the restriction of a to Go (which we also denote by a) is a quasicharacter of ( G o ,%,, y o ) , hence also a quasi-character of ( G o ,@,, p), relative to the group 6, of all left-translations of G o . According to Example 3.2.5,23 there exists a constant c, I c I = 1, such that

for almost all g E Go ; here, &(g) is the character of 8, induced by a, which is just the restriction to Go of the character of G induced by a. Consequently, if h is an arbitrary element of G , then, for almost everyg in the coset h-lG, , since hg E Go , we have .(g) = G(h-1) .(hg) = cE(h-1) d(hg) = cd(g).

I t follows that (3.2.4) holds for almost every g E G. ] Examples 3.2.5 and 3.2.6 show that the idea of a quasi-character generalizes the notion of a measurable character. We observe that, if (G, 8 ,p ) is a localizable measure space which is quasi-invariant under the transformation group 8, then any x’ E G. is continuous relative to the p-topology on 6. I n fact, given any h E 6, consider the operator U(h)defined in 93.1, 60. Then, for any appropriate choice of the vectors [, 7 €L2(G,23, p), we have2*

whence it is clear that x”(h)is continuous relative to the p-topology. We shall now make use of Lemma 3.1.12 to investigate the continuity of characters which are induced by quasi-characters. Let (G, 23, y) be a regular measure spacez5 which is quasi-invariant under a group of continuous measurable transformations 8 , and suppose that 6 is a topological group of the second category relative to an admissible topology. Then any quasi-character a of (G, 23, p ) (relative to 8 )induces a continuous character on 8.

Theorem 3.2.3.

as Translator’s note: To apply Example 3.2.5, one must make some additional assumption to ensure the measurability of the mapping (g,h) -+gh. 84 Translator’s note: Given any h, E 8, and any nonzero vector 6 E L2(G,9, p), we may choose I) = U(h,)(x[),thus ensuring that the denominator is nonzero in some sufficiently small p-neighborhood of h, Translator’s note: Again, we must require that p is not identically zero.

.

111.

144


PROOF. Given any positive number E, we partition the unit circle of the complex plane into a finite collection of disjoint Bore1 sets A, ,..., A, , such that the diameter of each A, is less than E. Accordingly, G is the union of the disjoint sets B,

= {g

I a(g) E A k } E 8, k

=

1,2,..., n.

(3.2.5)

Therefore, at least one of the sets (3.2.5), say B k o ,has positive measure. By the regularity of p, there exists a compact set K E 8 such that K C %k0 and p ( K ) > 0; in particular, if g, ,g, E K, then

I

-4gJl

< c.

(3.2.6)

Now, a(h-lg) is almost everywhere equal to a(g) Z(h-l), where is the character of 6 induced by a. Hence, by virtue of Lemma 3.1.12,28there is a neighborhood V of the identity in 6 such that, for any h E V, there exists an element g E K n hK satisfying (3.2.7)

a(h-1g) = .(g) Z(h-1).

Thus, if 12 E V, then it follows from (3.2.6) and (3.2.7) that

I d(h) - 1 I

=

1 1 - d(h-l)l

= I .(g) - .(h-lg)I

< t.

This shows that d is continuous at the identity of 6. Hence, by Lemma 3.2.2, 2 is continuous at all points of 6. ]

Corollary 3.2.4. Let G be a topological group, let 6 be a subgroup of G, and suppose that 6 itself is a topological group of the second category with respect to a topology which is stronger than that induced by G. Let (G, 8,p ) be a regular measure space which is left (or right) quasi-invariant under 6. Then, every quasi-character 01 of (G, 8,p ) (relative to 6) induces a continuous character on 6. In particular, every b-measurable character 01 of G, when restricted to 6, defines a continuous character on 6. Remark. As a particular case of Corollary 3.2.4, if G is a locally compact group, 23 is the u-ring generated by the compact subsets of G, and p is a left-invariant Haar measure, then every measurable character of G is continuous, and every quasi-character of (G, 8,p ) (relative to the left translation group G) induces a continuous character on G. I n fact, G is then a topological group of the second category (see Guan Translator’s note: Recall, however, the additional finiteness restrictions on p which were necessary to prove Lemma 3.1.12.


145

Zhao-zhi [I]), and (G, d,p) is invariant (hence afortiori quasi-invariant) under left translations. Applying Corollary 3.2.4, with 8 = G, the conclusion follows at once.

30 Topologies on Character Groups and Quasi-Character Groups I n the subsequent discussion, we shall require several kinds of topologies on character groups and quasi-character groups. We now proceed to describe these topologies. I. Let G be a group, and let H be a group consisting of certain characters of G. Let a. E H , let g , ,...,g, be any finite set of elements of G, and let E be any positive number. Form the set

For each ‘ Y ~ E H we, take the totality of sets of the above form as a neighborhood basis at a. . Thus, we obtain a topology on H , which we call the weak topology. This is, in fact, the weakest topology such that each g E G, when considered as a function a ( g ) , Q E H , is continuous on H . I t is easily seen that the weak topology makes H a topological group. and let H be a group consisting 11. Let G be a topological of certain characters of G. Let a. E H , let Q be any compact subset of G, and let E be any positive number. Form the set

For each a0 E H , we take the totality of sets of the above form as a neighborhood basis at a0 . Thus, we obtain a topology on H , which we call the strong topology; it may be described as the topology of uniform convergence on compact subsets of G. It is easily verified that the strong topology makes H a topological group. Since every finite subset of G is compact, it follows that every weak neighborhood2s in H is also a strong neighborhood, and consequently the strong topology is, in fact, stronger than the weak topology. On the other hand, there do exist topological groups G and corresponding groups of characters H such that the weak and strong topologies on H are actually different. 111. Again, let G be a topological group and let H be some group of Translator’s note: Actually, the definition given here is applicable even if the group operations are not continuous relative to the topology of G. 28 By a weak (strong) neighborhood, we mean, of course, a neighborhood in the weak (strong) topology.

111. GROUPS

146


characters on G. Let U be a fixed open set in G. Then, for each a. E H and positive number E, we form the set W(a0 ;

u, €1 = {a I sup I .(g) k-u

- 4g)l

< €1.

We take the totality of sets W(ao; U, E ) for all E > 0, as a neighborhood basis at ( y o . I n this manner, we obtain a topology T , on H ; we call T , the U-topology. Again, H is a topological group relative to this topology. Let Q be any compact subset of G. Then, there exist elements g, ,...,g, E G, such that (2 C x i = l g k U . Hence, if a satisfies

then

where g = gkh. Thus, we have

whence it is clear that, if the U-topology is stronger than the weak topology, then it is also stronger than the strong topology. Next, we shall consider certain topologies on groups of quasi-characters, or groups of measurable characters. IV. Let 9 = (G, 23,p ) be a measure space, and let L ( 9 ) denote the Banach space formed by the totality of integrable functions on 9. Let H be a family of bounded measurable functions on 9, functions which are equal almost everywhere being regarded as identical. (Two cases occur most frequently in the subsequent applications: either 9 is quasiinvariant under a transformation group 8 , and H is a group of quasicharacters on 9, relative to 8 , or G is a group and H is a group of %measurable characters on G.) Each element a of H defines a continuous linear functional on L ( 9 ) , as follows:

Thus, H may be imbedded29in the conjugate space of L(Q).I n this way, Translator’s note: In order that the mapping a -+ (., a) be injective, it is necessary and sufficient that Sa satisfy condition (i) of Definition 1.2.3. But in any case, we may still define the p-weak topology as the weakest topology on H such that the mapping a + (*, a) is continuous [taking the weak topology on the conjugate space of L(Q)].


147

the weak topology on the conjugate space of L(Q) induces a topology on H ; a neighborhood basis for this topology may be constructed as follows. Let a. be an arbitrary point of H , choose any finite set of functionsf, ,...,fn inL(SZ), take any positive number E , and form the set

We take the totality of sets of the above form as a neighborhood basis at the point a o . The topology obtained in this manner will be called the p-weak topology. If H is a group, then, in general, it does not necessarily follow that the p-weak topology makes H a topological group. Further on, however, we shall prove that, under certain circumstances, H does form a topological group relative to the p-weak topology, and we shall compare the strengths of this and other topologies. V. Let (G, 23, p ) be a measure space which is quasi-invariant under a transformation group 8 , and let H be some group of quasi-character~~~ of (G, 23, p ) relative to 8 (in particular, if G is a group, then H may be just a group of %-measurable characters on G). Take any set A E 23, p ( A ) < co, and consider the function 1/2

on H . Since

for all a , /3 E H , it follows that N A ( - )is a pseudonorm on H. We use the family of pseudonorms { N A ( . )to ) define a topology on H , as follows. Given any element a. E H , we choose an arbitrary set A E 23, p ( A ) < co, and an arbitrary positive number E , and form the set Y(a0 ; A , t) = { a I

NA(aail) < c}.

We then take the totality of such sets as a subbasis for the neighborhood system of a. . T h e topology so obtained will be called the p-topology. Using the fact that N A ( . )is a pseudonorm, it is easily verified that the p-topology makes H a topological group. If p is a finite measure, then the p-topology is equivalent to the topology induced by the metric NG(a/3-I).To show this, it suffices to prove that every set of the form Y ( a o ; A ,c) is open in the topology induced by NG(a/3-l).Take any ao In particular, if 8 contains only the identity transformation of G, then H is simply a multiplicative group of 8-measurable functions 01 which satisfy the condition I a(g)l = 1.

111.

148


Y(ao; A , E), and let E~ = E - N,(a,a;’); then, since N,(a) it is easily seen that Y ( a , ; G , €1) c W O ; A , 6).

a1 E

< Nc(a),

This shows that Y ( a o ; A, E ) is indeed an open set in the topology induced by the metric NG(aj9-l). T h e pseudonorm N , ( . ) may be used to introduce a pseudonorm on G,u. Choose any A E 23 such that 0 < p ( A ) < 03, and define a function N A ( q ) ,-q E Go@,as follows: N A ( ~=) inf NA(a). a€9

I t is easily verified that this is a pseudonorm on Go@. If F is a subgroup of Go”,and qo E F, we may take the totality of sets of the form

as a neighborhood basis at 70. In this manner, we obtain a topology on F, which we also call the p-topology. Relative to this topology, F forms a topological group. If p is a finite measure, we write NG(q)= N ( q ) , and in this case the p-topology is equivalent to the topology induced by the metric31 N(&l), 5, 7 E Gou. We now proceed to consider some properties of the topologies introduced above, and to give some examples.

Theorem 3.2.5. Let SZ = (G, b,p ) be a measure space which is quasi-invariant under a transformation group 6, and let H be some group of quasi-characters of relative to 6. Then, the p-topology on H is stronger than the p-weak topology. PROOF. Given any a. E H and any p-weak neighborhood X(a0 ;f1

,...,fn , 4,

we need only prove that there exists a set A a positive number such that Y(a0 ; A , €1)

Now, since such that

fk

E

c q u o ;f1 ,...,

fn

23, 0 < p ( A ) < 9

CO,

and

4.

E L ( G ,8,p ) , there exists a set A E b, 0 < p ( A ) < 00,

81 Since the subgroup fn is closed with respect to the metric N(a/3-’), a, /3 E G’, it follows that N([q-l), 5, q E Gou,is indeed a metric on Gou.In fact, Gou, with the metric N ( [ q - l ) , [, q E Gou$is just a quotient metric space of Gou(relative to the metric N~(ap-’), a,/3 E G”).


149

and such that sup Ifk(g)I
0 and A E 8, 0 < p ( A ) < CO. By the regularity of p, there is a compact set Q E 23 such that Q C A and €2

P(A - Q ) < 8 *

Choose

Then, if

01

JA

E

V(ao ; Q , el), we have

I .(g)

- "o(g)I2 d P k )

€2

€2

2

2

that is, NA(aa;')

< z.

be a fundamental sequence in Gu, that is,

as m, n -+ co. Since the space L2(52)is complete, there exists a quadratically integrable function a such that

furthermore, we can select a subsequence {an,(g)} which converges to 01 almost everywhere. Therefore, since I a,,(g)( = 1, we may choose 01 such that 1 a(g)J= 1. Moreover, for any h E 6, it follows from the quasi-invariance of p that

holds for almost all g E G, that is, (Y is a quasi-character. And, by (3.2.8), limn+mNG(~na-l) = 0. Hence, Gu is complete. ] Remark. If p is not finite, but is localizable, then it can be proved that the multiply pseudometric space (for the definition, see Guan Zhao-zhi defined by the family of pseudometrics {NA(aP-l)} is sequentially complete.

Corollary 3.2.8. Under the hypotheses of Theorem 3.2.7, the group Gouis complete relative to the metric N(&-I), (, q E Gou. PROOF. Let (8,) be a fundamental sequence in Gou, that is, limm,n+mN((;l(,) = 0. Then, we can select a subsequence {(,}, k = 1, 2,..., such that N(,$;~_,f,,)< 1 / 2 k . For each k 2 2, choose an element ak E t;:-,(,, such that N(ak) < 1/2k, choose any element k 0 1 ~E ( ,, , and form the product f l k = a l. Then, obviously f l k E ( , , 1 and when 1 > k, we have flklfll = f l v = k + l a , , hence

nl=l

1

Wiglcligd
0, it follows that a(h) = 1; consequently, both sides of (3.2.9) vanish. On the other hand, if JA p ( g ) dp(g) > 0, then, since V C V, , (3.1.17) holds, Thus, in either case, we have

where c’ = c(p(A))ll2.Let 7 = a% be the residue class containing a. Taking the supremum of the right-hand side of (3.2.12) over all OL in q, we get sup I Lqh) - 1 I < C ’ N A ( 7 ) . heV

Hence, given any point qo E Gou, and any V-neighborhood

of the point 2,

=

Sv0,there exists a p-neighborhood 2(qo ; A , E/c’) of

rl0 such that SZ(7, ; A , E/C‘)

c {a I sup I a@)- a,(h)l < hEV

which shows that S is continuous.

€},

]

Theorem 3.2.10. Let (G, 23,p) be a regular measure space which is strongly quasi-invariant and ergodic with respect to a group 6 of continuous measurable transformations. Suppose that 6 has an admissible topology which makes (5 a connected topological group of the second category. Choose any nonzero element z/ of L(G, 23,p ) such that +(g) 2 0, g E G. Form the convex function

Then, the topology on Gu induced by Nd is independent of the choice of +, and is, in fact, equivalent to the p-topology.


153

PROOF. We shall first prove that, given any sequence {a,] C Gu such that limn+mN,(a,) = 0, there exists a subsequence {a,,} which converges to 1 almost everywhere on G. Choose a positive number a, sufficiently small so that the set A = (g I +(g) > a> has finite positive measure. Then, by virtue of the inequality

we know that {a,} converges in measure to I on the set A, hence, there exists a subsequence {an,) which converges to 1 almost everywhere on A. Consider the set E = {g I lim an,(g) = 1). n +n

Since A is contained in E (modulo a null set), we have p ( E ) > 0. We shall prove that p(G - E ) = 0. In view of the ergodicity of p with respect to 6, it suffices to prove that E is a quasi-ergodic set. Now, by the regularity of p, there is a compact set K E 8,0 < p ( K ) , such that K C E. According to Lemma 3.1.12,34 there is a neighborhood V of the identity in 6, such that p ( K n hK) > 0

for all h E V . Therefore, for each h E V , there is an element g E K n hK, such that cin,(h-lg) = En++)

holds for all n'. But g E K n hK implies that g follows that lim

n"m

&%,(A)

(3.2.13)

a,,(g) E

E, h-lg

=1

E

E, hence, it (3.2.14)

for every h E V. Since d is a character of 6, the set 6, = { h I n'+m lim En,(h) = l }

is a subgroup of 6, and, by (3.2.14), we have 6,3 V . By virtue of the connectedness of 8 , it follows that 6, = 6, that is, limn,,, d,,(h) = 1 for all h E 6. On the other hand, for each h E 6, there is a null set Ghin G such that (3.2.13) holds for all n', provided g E Gh . Consequently, s4 Translator's note: Again, recall the qualification concerning the validity of Lemma 3.1.12.

154

111.


g E hE - G, implies that g E E ; thus, hE - E is a null set. This shows that E is indeed a quasi-invariant set, and therefore

lim a,#(g)

n’+m

=

1

(3.2.15)

holds for almost all g E G. Hence, for any set A E 23, with p ( A ) < a, it follows by the Lebesgue dominated convergence theorem that lim NA(a,,) = 0.

n‘+m

From this, we easily deduce that the topology induced by N&is stronger than the p-topology. Conversely, given any positive number E , we choose a set A E 23, 0 < p ( A ) < co,such that

and c = sup I#(g)l &!€A

Then, for any a

E {a

< 00.

I N A ( a )< r / 2 dc},we have

that is, { a I N A ( a )< ~ / dc} 2 C { a I N,(a) < E ) . Therefore, the p-topology is stronger than the topology induced by N 4 . Thus, the topology induced by N , is identical with the p-topology, and is therefore independent of the choice of 4. 3 T h e above theorem may be used as a criterion for the equivalence of two measures. I n this connection, see Theorem 4.2.14 (which is an analog of Theorem 3.2.10) and Corollary 4.2.15.

Theorem 3.2.11. Let l2 = (G, B,p) be a localizable measure space which is quasi-invariant under a transformation group 6. Suppose that 8 has been given a topology which is stronger than the p-topology. Let 6~be given the strong topology, and let Gu be given the p-weak topology. Then, the mapping cx 8, from GUto @, is continuous. PROOF. First, we shall prove that, for any fixed ~ E L ( Q )G’ , x 6, the function --f

da,h ) =

a(hg)f(g)449,

(a9

h) E GU x 6,


of a l , then

1~

( ah), - v ( a l , hl)l

v(01,h). Now, given any

0 1 ~E

< 2.5. This

155

proves the continuity of

GF,we choose an f eL(SZ)such that

Also, we choose a sufficiently small neighborhood V of

J 4 g ) f ( g )44g) + 0 for all a E V. Thus, when

01

E

V , the character

, such that

0 1 ~

156

111. GROUPS


is a continuous function of ( a , h). I t follows that, for any positive number and any compact set Q in 6, the inverse image (under the mapping a --t a) of the strong neighborhood

E

(of d, E @) is open in Gu. From this, one easily deduces that the mapping a -+ 6 is continuous. ]

Corollary 3.2.12. Let G be a locally compact topological group, and (G, 8,p ) a left-invariant Haar measure space. Then, the p-weak topology and the strong topology on G* are identical. PROOF. By Lemma 3.1.29, the original topology of G is stronger than the p-topology. Following the argument used in the proof of Theorem 3.2.11, it is easily seen that the p-weak topology on G* is stronger than the strong topology. On the other hand, it follows from Theorems 3.2.5 and 3.2.6 that the strong topology on G* is stronger than the p-weak topology. Hence, these two topologies are identical. 3 T h e next corollary is an immediate consequence of Theorems 3.2.5 and 3.2.1 1. Corollary 3.2.13. Under the hypotheses of Theorem 3.2.11, if GY is given the p-topology, then the mapping a + d is also continuous. Note, however, that the conclusion of Corollary 3.2.13 is weaker than that of Theorem 3.2.9. Theorem 3.2.14. Let 52 = (G, 8,p ) be a localizable measure space which is quasi-invariant and ergodic with respect to a transformation group 6. Let $3 = {ca I a E Gu,0 c I}. Then $3 is compact with respect to the p-weak topology. PROOF. By Theorem 2.4.14, the conjugate space ofL(S2) is isomorphic to L"(52), the totality of essentially bounded measurable functions on 52. Moreover, since the unit ball in the conjugate space of L(52) is compact in the weak topology, the same is true of the unit ball B in L"(52). Hence, we need only prove that $3 is a weakly closed subset of B, and it will follow that $ itself is weakly compact. Let {a, , X E A } be a generalized sequence in $3 which converges weakly to an element a~ B . Since 0 ~ 6 we , need only consider the case a # 0, and so we may as well suppose that 01, # 0 for all h E A . If 01, = c,&, where 0 < C, 1 and /3, E Gu, then we write L?, = PA . We shall now prove that, for every fixed h E Q, the generalized sequence of complex numbers {a,(h), h E A } is fundamental.

<
P‘*(H, n G’) is obvious, but the reverse inequality is not.

3.3. Integral Representation of Positive DeJinite Functions

177

hence P’*(H, n G‘) = I , and therefore P’*(GB) = 1. For every A E g‘, define P B ( An GB) = P’(A). (3.3.33) Since gB = { A n GB I A E f!} and P’*(GB) = 1 , it follows again from $1.1, lo, that P B is a probability measure on (GB,gB),moreover, it is easily seen that

holds for every g E G. This, combined with (3.3.15), shows that (3.3.19) holds for all g E G. (6) It only remains to show the uniqueness of the measure PB. Suppose that PIBis another measure on (GB,gB) satisfying (3.3.19). Define a probability measure P,’ on (G’, b’),as follows: P,’(A)

=

PIB(A n GB),

A

E

3’.

(3.3.34)

It then follows from (3.3.19) and (3.3.34) that PI‘ satisfies condition (3.3.15). But, by Theorem 3.3.4, this measure is unique, that is, P,’ = P’. Hence, from (3.3.33) and (3.3.34), we conclude that P I B = PB- 1 We point out that, in Theorem 3.3.5, the condition that {%f(q:’(g)-Ig)) converges to 1 almost everywhere (relative to p ) cannot be dispensed with. This may be seen from the following example. Example 3.3.1. Let R denote the additive group of all real numbers, b the totality of Lebesgue measurable subsets of R, and p any probability measure equivalent to Lebesgue measure. It was shown in Example 3.2.4 that R B is just the totality of functions of the form at(.)

= eit”,

x E R,

hence the right-hand side of (3.3.19), in this case, takes the form (3.3.35)

where P is a probability measure on ( R , 23). It is easily seen that, if R is given the ordinary Euclidean topology, then (3.3.35) must be a continuous function on R . Consider the function f(x) =

1,

lo,

x =

x #

0, 0.

111.

178


Obviously, f is a %-measurable positive definite function on R. But f is not continuous, and hence cannot be expressed in the form (3.3.19).

Lemma 3.3.6, Let G be a topological group satisfying the first axiom of countability, let 23 be a o-algebra in G which contains all the open subsets of G, and let (G, 9, p ) be a regular locally finite measure space. Then, there exists a partition {E, I U E C} of G, and a corresponding family of mappings {vk) I o E Z, n = 1, 2, 3, ...}, satisfying the conditions (i), (ii), and (iii) (preceding Lemma 3.3.5) such that, for every D E C, lim &)(g) n+ m

=g

(3.3.36)

holds for almost all g E E, . PROOF. Let U be a family of pairwise disjoint subsets E of G, satisfying the following conditions: E : E E 23, 0 < p ( E ) < 03, and, for any open set V , either E n V is empty or p( V n E ) > 0. Let 8 denote the totality of such families U, and let ?be j partially ordered by inclusion, that is, U, < U2 means that U, C U2 . Using Zorn’s lemma, it is easily seen that 8 has a maximal element,46which we denote by {E, 1 o E C}. We assert that any compact set K intersects at most countably many E, . I n fact, it follows easily from the local finiteness of p that there exists an open set V , having finite measure, such that V 3 K. Therefore, since

there are at most countably many E, such that p ( E , n V ) # 0, that is, at most countably many E, such that E, n V is nonempty. Hence, a fortiori, there are at most countably many E, such that E, n K is nonempty. Let R = E, . We shall prove that G - R is a null set. T o do this, it suffices to prove that, if K is an arbitrary totally bounded set, then K n (G - R ) is a null sets4’ Since K intersects at most countably many E, , we have K r\ R E 23, hence K n (G - R)E 9. Suppose that

u,

48 Translator’s note: It is not immediately clear that 5 is nonempty, unless one makes some additional assumption, for example, that every nonempty open set has positive measure. 47 Translator’s note: The justification for this assertion is obscure. The crucial point is proving that R E 8,and it is difficult to see how this follows from the argument given here. Of course, if p is o-finite, then there are at most countably many sets E, In that case, R obviously belongs to 23, and one can simply consider an arbitrary compact set K C G - R; the proof can then be completed in a manner similar to that indicated in the text.

.

3.3. Integral Representation of Positive Dejinite Functions

179

the measure of K n (G - R ) were not zero; then K n (G - R ) must contain a totally bounded set E having positive measure. Obviously, the intersection E n E, is empty for every cr E 2'. Let U , 3 U , 3 3 U , 3 be a neighborhood basis at the identity e of G; we may assume that p(U,) < cx). Since E is totally bounded, there exists, for any n, a finite set of points g p ) ,...,ggn) in E such that u 2 1 g ! n ) U ,3 E. It may happen that E n (gy'"'U,) is a null set for certain indices n, v ; let F be the union of all (at most countably many) such sets, and let E, = E - F. Then E, E B, Eo n E, is empty for every u E Z, and 0 < p ( E o ) = p ( E ) < co. We assert that, if V is any open set which intersects E, , then p ( E , n V ) > 0. I n fact, suppose g E E, n V ; since g is an interior point of V , there is an n such that g U , C V , and, for any m, since g E E, , there is a v such that g E g ~ " ) U , , that is, g-lg;"' E U;'. Hence, if m is sufficiently large, then U;' UrnC U , , and we have 1..

gj""'um CgU&lUmc v. But g E ~ ; " ) U,,g E E, , hence E n (g:"'U,) is not a null set [since E, was obtained from E by removing all those sets E n gSm'Um such that p ( E n gjm)Um)= 01. Since E, n V 3 E, n (gbm)Um),this implies that E, n V is not a null set. Thus, {E, , u E Z} u (E,} E 3.This contradicts the maximality of {EDI D E Z}. Therefore, we conclude that G - R is a null set. We may assume that G = R (for otherwise, we could incorporate G - H in one of the sets E,). Thus, {E, I u E Z} is a partition48 of G. Next, we proceed to construct the mappings {+}I.: Clearly, we may assume that U;lU, C Urn-,, n = 2, 3 ,... . Now, for each n, n = 1, 2,..., and each U E Z ,one can choose a finite or countable set of elements E E, , v = 1 , 2,..., such that p

(E,

-

u "

hFACT,) = 0.

(3.3.37)

I n fact, by regularity, E, is the union of countably many compact sets and a null set, and each of these compact sets can be covered by finitely many sets of the form hU, , where h E E, , hence, there does exist a finite or countable set (hi:;} C B such that (3.3.7) holds. Since h,!:AU, E 23, all the sets EbYL

**

=

(h?kUn

u

V-1

-

U n ) n E,

k=1

Translator's note: Again, the proof of this assertion is not obvious (unless p is o-finite).

111. GROUPS

180


also belong to 23,and if u # u', then E,!:Aand E$',L are disjoint. Moreover, if we let EkL = E, - E,!::,

u v

then, by (3.3.37), we get p(EkL) = 0.

(3.3.38)

We shall define the mappings {c#} by induction. First, define &) as follows. For each nonempty set E::: , choose any element g::! E E,!:] , and let q$)(g) = g : ] for all g E E::! . Clearly, &) satisfies conditions (i) and (ii). Now, assume that mappings &), ..., ypl have already been defined, satisfying conditions (i), (ii) and (iii) (for 1 n m I - I). Given any u, we use the letter s to denote all the indices such that let #(g) = T'&(g). If the set gEi-l E E,!:; . If g E E$' n

< <
,

6 E e,

'p E e k 2 ( Q .

(3.4.6)

Then, we say that (0,G) is a dual of (Q, 6) (or that 0 is dual to a) and that F is an associated Fourier transform (from L2(Q)to Q k 2 ( Q ) ) ; we describe such a mapping F, in general, as an L2-Fourier transform. First, we prove the following existence theorem.

Theorem 3.4.3. Let Q = (G, 23,p) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transof (Q, 6) and an formation group 6. Then, there exists a dual (0,G) associated Fourier transform from L2(Q)to f!k2(0). PROOF. Let be the totality of symmetric multiplicative linear functionals on %(Q, 6), let !8 be the totality of Bore1 sets in %, and, for each h E 6, let h(g) = U(h)(g). Obviously, the function h(.) is ?%measurable, and I h(f)l = 1 for all E Since % has uniform multiplicity K, we know by Corollary 2.4.7 that there exists a measure

e.

192

111.

GROUPS W I T H QUASI-INVARIANT MEASURES

(e,

space 0 = @, $) and a unitary mapping F from L2(sZ)onto such that, for every A E a, FAF-W)

=

A(&

m),

g E e, f

Qk2(0)

E Qb2(@,

and, in particular, (3.4.6) holds. Let 8 be the linear hull of the family of functions & = { h ( j ) I h E Q}. If 8 be regarded as a set of multiplication operators on 2k2(Q), then F'2IF-l is just the closure of 3 in the strong topology. Now, take any A , € % , and let E be any a-finite set in 0; let E be expressed as the union of a countable family of disjoint sets {Ei) C @, with /2(Ei) < co, i = I, 2,... . If 4, is any unit vector in k-dimensional Hilbert space, then the vector-valued function f@) =

c

2-jl"(Ej)-'12

cEi(g)fO *

9

tE&'

j

[the summation being extended over those indices j for which ek(0).Hence, for every natural number n, there is a function f,(j) E 8 such that

$(Ej) > 01, belongs to

Therefore, using a diagonal process, one may select a subsequence of {f,} such that f,@) -+A,(j) for almost every j E E. Since {fn,> C 8, and since @ is, by definition, determined by the totality of functions A,@), A, E a,it follows easily that 8 is a determining set for 0,whence & is, afortiori, a determining set for 0. Let &i'(L2(Q))denote the totality of bounded linear operators on L2(Q), and, for each x E Gu, define a transformation T, : B --f V ( x )BV(x)-', B ~ a ( L ~ ( l 2 Obviously, )). T, is an automorphism of a(L2(sZ)),and TxB* = (TxB)* for every B E G?(L2(sZ)).Moreover, if h E 6, then, by (3.4.1), we have T,U(h) = 2(h) U(h). Consequently, T , induces an automorphism of a(Q, Q), and this automorphism depends only upon 2. Define (@')(A)= @(T,A), A E %(Q, 6), d E &'.

{f,.}

It is easily verified that $2 E (?,that the mapping2 --t &j is @-measurable, is an isomorphism of 6% onto that the correspondence 2-4 65 = {$I x E Gu), and that h($) = h($j)/h(j)= 2(h) for all j E e. Thus, it only remains to prove that $ is quasi-invariant under Q.

s Write r ( x ) = FV(x)F-l. Obviously, r ( x ) is a unitary operator on Sk2(fi), and, by (3.4.1), (3.4.7)

q h ) O(h) V ( x ) = V ( x ) O(h)

therefore, for any f E

5, w f ( i ((6) > = f(@)

Write that

c(g)for P(x) ((2);

w &i)-

(3.4.8)

since P(x) is unitary, it follows from (3.4.8)

Now, the two set functions PlW

=

J

II t(9)l12d&(i),

PZP) =

J

E

II f(@9)l12

d@i(i)

are finite measures on (&', 8).Since & is a determining set for fi, it is clear by Lemma 1.1.669that 5 is dense in L2((?,8,p1 p2). Hence, any bounded measurable function on &' may be simultaneously approximated in L2((?,8,p l ) and L2(&',8,p2) by a sequence of functions {fn} C 5.Therefore, it follows from (3.4.9) that

+

for any bounded measurable function f.In particular, choosingf = C, , the characteristic function of an arbitrary set E E 8,we get p 2 ( E )= p l ( E ) , that is,

J

II 5(i)l12 d & ( i ) = J II "1i)l1 d @2 i(i)

(3.4.10)

68 Translator's note: The author's proof of Lemma 1.1.6 is valid only for the real case. However, it is a simple exercise to show that the result is also valid for the complex case, provided that the algebra 3 is invariant under complex conjugation, which condition is obviously satisfied by 5 in the present situation.

111.

194


for any E E 8.If @ ( k l E )= 0, then the right-hand side of (3.4.10) is zero, hence the left-hand side is zero for all 5 E 2k2(fi), and it follows easily that @ ( E )= 0. Since 2 is an arbitrary element of &, this shows that fi is quasi-invariant under 6. Ia0 Using the proof of Theorem 3.4.3, together with Corollary 3.3.2, one can also obtain the following result.

Corollary 3.4.4, Let Q = (G, 8,y) be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 6. Let S denote the p-topology on 6, and let (? be a spectral groupa1 for (6,S) such that @ C (?. Let 8 be the totality of weak Bore1 sets in and take 0 = 6s. Then, there exists a &quasiinvariant measure @ on @), such that (((?, @, @),&) is a dual of (SZ, 6) with respect to an appropriate Fourier transform from L2(Q)to

c,

e k 2 G

(e,

B, 6).

Next, we proceed to find the general form of all L2-Fourier transforms associated with a fixed dual. Let U, = { U(2) 1 9 E G} be the transformation group in L2(fi) corresponding to 6. We may also regard U, as a group of transformations of ek2(0), namely, if 5 E gk2(Q),4 E 6, we define (3.4.11)

Clearly, U(2)is a unitary operator in Qk2(0). We note that the group of multiplication operators on Qk2(Q), corresponding to @, is just { o ( h ) 1 h E S}[see (3.4.6)]. A trivial calculation yields the commutation relation E(h) U(&)O(h) = O(h) U(&), h E 6, & E 0, (3.4.12)

v(a)

analogous to (3.4.1). For any (Y E GuI let W ( a )= U(&)]where aZ denotes the image under the mapping (3.4.3) of the character G induced by a ; thus, V(a) = W ( a )U(&-l), a E Gu. (3.4.13) 6 o Translator’s note: Since the mapping h -+ h(2) defined in this proof is the composition of the mappings h U(h)and U(h) + U(h)(&,and since h --* U(h)may not be injective, it is obviously impossible to prove that h h ( . ) is injective, unless further conditions are imposed. This suggests that the requirement of Definition 3.4.3 [that (3.4.4) be injective] is too strong, and that one should, instead, require the existence of an appropriate isomorphism U(h) + h(.), from U into @. Translator’s note: T h e term “spectral group” was defined with reference to a topological group. In the absence of further assumptions, (8, 2)may not be a topological group in the usual sense, since there is nothing to guarantee that the p-topology is --f

Hausdorff.

-.

3.4. L2-Fourier Transforms

195

We see from (3.4.7) and (3.4.12) that W(a)and ??(h) commute, that is, W(a)O(h) = O(h) W(a),

01

E Gu.

(3.4.14)

Since { O(h),h E 0}generates the multiplication algebra %Ilkover Qk2(L?), it follows from (3.4.14) that W(a)E { o ( h ) ,h E S]’= %Rk’.If we assume that k N o and that $ is localizable, then, by Lemma 2.4.20, we know that there exists a measurable k-dimensional unitary operator-valued function z(6; a),6 E such that

1 on some compact neighborhood N in G*. Hence, by (3.4.40),we get

Thus, p' is locally finite, and we conclude that p' is a Haar measure. Finally, using (3.4.40),it is easily shown that F, is the required L2-Fourier transform. ]

Example 3.4.5.

Let 1 = 1 , 2,..., be a finite or countable set of

74 Translator's note: Although these assertions seem highly plausible, more detailed arguments would be welcome.

111. GROUPS

204


indices, and for each I, let Q, = (G, , 8 , ,p J be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 6 , . For any finite sequence h = (h, ,..., hn), where h , E 8 i ,I = l,,.., n, we define a transformation by hg

= @,g,

Y...,

hng, ,gn+,

,*..I,

where g = {g, ,...,g, ,g,+, ,...} is any element of X I G , such that g,E 'D(h,), I = 1,2,..., n. Let 6 denote the totality of such transformations h, and suppose that Q, is a probability measure space for all but a finite number of indices 1. Then, the product measure space Q = x, Q, is quasi-invariant under and cyclic of order k, relative to 8.Moreover, if, for each I , (0, , 6,)is a dual of (Q, , 6,),if 0,is a probability measure space for almost all I, and if we define a corresponding transformation group 6 in the same way that 6 was defined above, then (X, 0,, 6) is a dual of (Q, 6). The proof is left to the reader.

n,

30 Duals of Ergodic Measure Spaces Lemma 3.4.9. Let Q = (G, b,p ) be a localizable measure space which is quasi-invariant under a transformation group 6. Let B E b, and let HB be the closed linear subspace of ek2(Q) formed by the totality which vanish almost everywhere outside of B. of functions in ekz(Q) Then, B is quasi-invariant under 8 if and only if H B is invariant under the group of unitary operators U = { U(h) I h E S}. PROOF. Assume that B is quasi-invariant under 6, and let y E H , . Then, for any h E 0,it is clear that y(hg) is zero for almost all g E G - B, hence

for almost all g E G - B, that is U(h)y E H E . Thus, H E is invariant under U. Conversely, let B € 8 ,and assume that H B is invariant under U. If B were not quasi-invariant, then there would exist an element h E 6 such that hB - B is not a p-null set, therefore, since Q is localizable, one could find a set E E b such that 0 < p ( E ) < 00, 0 < p(hE) < 00, E C B , hE C hB - B. Then, if C, denotes the characteristic function of E, and y o is any fixed nonzero vector in k-dimensional Hilbert space, the function y E ( . )= C,(.) y o would belong to H , , and

205

3.4. L2-Fourier Transforms

Thus, ( U ( h )cpE)(g) # 0 for allg E hE C G - B, and hence U(h)cpE E H E . But this contradicts the invariance of HB under U. We conclude that B is quasi-invariant under 8. 3

Corollary 3.4.10. Under the hypotheses of Lemma 3.4.9, let M be the smallest weakly closed operator ring containing both the multiplication algebra W(9) over L2(L?)and the group U = { U(h) I h E S}. Then, 9 is ergodic with respect to 8 if and only if M is a factor; SZ is weakly ergodic with respect to 6 if and only if the center of M contains no countably decomposable projection 0perators.~5

Theorem 3.4.11. Let 9 be a localizable measure space which is quasi-invariant under and cyclic of order k relative to a transformation group 8.Suppose that SZ is normal with respect to 8,and let (a,6) be a dual of (L?, 6).If L? is ergodic with respect to 6, then is ergodic with respect to 6. PROOF. Assume that is not ergodic with respect to 6, and let 8 be a set in @ which is quasi-invariant under 6, with @(I?) # 0, - 8)# 0. Let H g denote the closed linear subspace of Qk2(fi) generated by the totality of vector-valued functions which vanish on - 8.Then, (3.4.42) (0) f HE # 2&9.

a

&(e

Moreover, by Lemma 3.4.9, H g is invariant under all the operators U(x), x E 6, and obviously Hg is also invariant under every operator of the form 5 4z ( * ;a) ((-). Since, by (3.4.18), P(a) = x(*;a ) U(S-l), a E G@,it follows that HEis invariant under all the operators 01 E GW. It is also obvious that HE is invariant under the operators t)(h), h E 8. Write H = F-lHg. Then, H is a closed linear subspace of L 2 ( 9 ) which is invariant under the groups U and 23. Let P denote the projection operator from L2(9)onto H ; we then have P E 23’. Since G@is a determining set of functions on 9, the group 23 generates the multiplication algebra %I over L2(sZ),and since 9 is localizable, !Ul is maximal commutative weakly closed. Hence, 23’ = !Ul’ = W, so that P EW. Let M be the weakly closed operator algebra generated by U u %I; since P E U’, it follows that P E M n M’. But by Corollary 3.4.10, M is a factor, hence P = 0 or P = I , which contradicts (3.4.42). We conclude that fi is ergodic with respect to 6. ] In general, the converse of Theorem 3.4.11 is not valid, as is shown by the following example.

r(01),

76 Translator’s note: This is clearly not a necessary condition for weak ergodicity, since it is obviously not satisfied when L2(Q)is separable, and examples of weakly ergodic o-finite measure spaces Q, such that L2(Q)is separable, are easily constructed.

111. GROUPS

206


Example 3.4.6. Let G be a group of just two elements {a, e}, where a # e, a2 = e, let b be the totality of subsets of G, and let p be the measure on 23 defined by &(a, e}) = 2, ~ ( ( u }= ) p((e)) = 1, p(empty set) = 0. Let 6 be the subgroup of G consisting of just the unit element e, and let (5 be the totality of functions from G to the complex numbers of unit modulus. Then, (G, 8 ,p ) is a finite measure space which is invariant under and cyclic of order 2 relative to the translation group 6, and is obviously normal with respect to 6. However, p is not ergodic with respect to 6, in fact, both of the singletons {e} and {u} are nontrivial quasi-invariant sets. Since the quasi-characters (3 induce just the one character 1 on 6, we see that (G, b,p ) has a dual measure space (&, B, $) defined as follows: & is a group containing only a single = 1, element 6, b consists of {6} and the empty set, $(empty set) = 0, and 6 consists of just the identity transformation. Obviously, (&, 8,$) is ergodic with respect to 6. This shows that a quasi-invariant measure space which is not ergodic may have an ergodic dual. However, when k = 1, the converse of Theorem 3.4.1 1 is, in fact, valid.

a({$})

Theorem 3.4.12. Let Q be a localizable measure space which is quasi-invariant under and cyclic relative to a transformation group 6. Suppose that SZ is normal with respect to 6, and let (0, 6) be a dual of (in, 6).If a is localizable and ergodic with respect to 6, then SZ is ergodic with respect to 6. PROOF. We follow the argument used in proving Theorem 3.4.1 1. Assume that SZ is not ergodic, and let B E B be quasi-invariant under 6, with p ( B ) # 0, p(G - B ) # 0. Let HB denote the totality of functions in L2(SZ)which vanish almost everywhere in G - B. Then, H , is a closed linear subspace of L2(Q) which is invariant under U and 2?, moreover, (0)#

HB

#L2(Q).

(3.4.43)

Let $5 = FHB ; then, $jis a closed linear subspace of Cl2((a) = L2(sfi) which is invariant under the groups 0 = { o ( h ) I h E 6) and 6= I 01 E Gu}. Let Q be the projection operator from L2(0)onto $5, and let $I denote the multiplication algebra over L2(sfi).Now, according to Definition 3.4.3, & is a determining set of functions on sfi, hence, it follows by the results of $2.4 that generates '$However, I. since 0 is localizable, $I is maximal commutative weakly closed. Thus, we have Q E a' = '3%'= $I. If 01 E Ge, and z(g; a ) is the function appearing in (3.4.18) (since k = 1, this is presently a numerical-valued function),

(r(01)

207

3.4. La-Fourier Transforms

then the operator .$ +z ( * ;a ) belongs to @. Therefore, since U(4-l) = z ( * ;a)-l P ( a ) , we see that 5 is invariant under { U ( x ) I x E 6}, and hence Q E { U(x) I x E &}'. Let i?! denote the weakly closed operator algebra generated by@ and { U ( x ) I x E 6}.By virtue of Corollary 3.4.10, the ergodicity of b implies that it? is a factor. But Q E i@ and Q E A?', hence Q is either 0 or I , which contradicts (3.4.43).We conclude that SZ is ergodic with respect to (ti. 3 ((a)

40 Strongly kth Order Cyclic Measures We shall now consider, in particular, the case where the dual is a finite measure space. Since every a-finite measure is equivalent to a finite measure space, some of the following results may easily be extended to a-finite measure spaces.

Definition 3.4.4. Let 9 = (G, 23, p ) be a localizable measure space which is quasi-invariant under a transformation group (ti. Let cp E L2(SZ), and let H, be the smallest closed linear subspace of L2(9) which contains cp and is invariant under all the operators U(h), h E (ti. We call H, the subspace generated by the cyclic element cp (and the group U). Suppose that {cpA 1 h E A} is a family of vectors in L2(52) such that

and that the function

is independent of A. Then, we say that { y hI h E A} is a 6-cyclic family of elements, that +(h) is the corresponding adjoint function, and that (G, 23, p ) is strongly kth order cyclic relative to 6 , where k is the cardinal number7s of A .

Theorem 3.4.13. Let 9 = (G, 23,p ) be a localizable measure space which is quasi-invariant under a transformation group 6 .If Q is strongly Kth order cyclic relative to (ti, then l2 is cyclic of order k relative to 6. Conversely, if L 2 ( 9 )is separable, and 9 is cyclic of order k relative to (ti, then k K O , and 9 is strongly kth order cyclic relative to (ti. PROOF. Suppose there exists a 6-cyclic family of elements { y AI h E A}, where the cardinal number of A is k. Let % denote the operator algebra corresponding to the transformation group (ti. Then each HeA is an

O}; then, E is a-finite relative to p‘, hence also o-finite relative to p. By virtue of the (weak) equivalence of p’ and p, there exists a measurable function

When p’ and p are strongly equivalent, G,, and G,, coincide.

4.2. Linear and Quasi-Linear Functionals

247

such that g, = #'(dp'/dp) €L1(L?), y 2 0, g, does not vanish almost everywhere, and

But according to Theorem 4.2.14, R, and R, induce the same topology. Therefore, R, and Ri, induce the same topology. ] Corollary 4.2.15 may also be used to establish the inequivalence of two given measures.

Example 4,2.1. We again consider the measure space described in Example 4.1.2. Let It denote the totality of linear functionals on I having the form

for some natural number n and some set of real numbers fi ,...,f,(if we take the product topology 2 on 1, then It is just the totality of continuous linear functionals on I ) . Let 1, denote the linear subspace of I consisting of all sequences of the form {fl ,..., fn , 0, 0,...}. Then, the correspondence f -+{fl ,...,f, ,...} is a one-to-one mapping of It onto lo . Also, since every f in It is a measurable linear functional on ( I , S U ) we , have It C G, . We now consider the following special case. If 1 a 2, the function

<
> E E, j c sj> will be called the Borel cylinder with base E corresponding to {vl ,..., y,}. If the elements y1 ,...,cpn generate the linear subspace M of 0,then we also call the above set a Borel cylinder corresponding to M, or a Borel M-cylinder. T h e totality of Borel cylinders corresponding to a fixed M forms a cr-algebra, which we denote by S ( M ) [or S(y, ,..., 4 1 , and the totality of all Borel cylinders forms an algebra S. Let 23 denote the smallest a-algebra containing S; we call the elements of 23 weak Borel sets corresponding to @, or @-weak Borel sets. I n particular, if @ is a linear topological space and sj C Ot, then we simply refer to the 23) is elements of 23 as weak Borel sets. In any case, it is clear that ($j, a linear measurable space. Let @ be a linear space and 8 a linear subspace of @*.Let 5 be the smallest a-algebra of subsets of @ which contains all sets of the form

{v IF(v) < a}, The elements of

5 will be

-co

< a < co, F E B .

called weak Borel sets corresponding to

6,or


249

!+weak Borel sets. I n particular, if @ is a linear topological space, and $j C Ot,then the elements of 8 will simply be called weak Borel sets. If fi distinguishesz7 the points of @, then we may imbed @ in 5” by identifying each point p E @ with the element p1E $indefined by

Vl(f)

=

f(vL

f E

5.

Let = {pl I p E @}. Then the weak Borel sets in 5 corresponding to are just the weak Borel sets corresponding to @. Thus, in this case, the two concepts of “weak Borel set” described above essentially coincide. T h e following lemma shows that the weak Borel sets constitute a sufficiently wide class of sets. Q1

Lemma 4.2.16.28 If @ is a separable countably normed space, then every open (or closed) subset of @ is a weak Borel set. PROOF. Let

/I VJ Ill

< II v < ..- e /I F Iln e 112

*--

1.

Thus, g, E E. This proves that S , is a weak Borel set. I n a similar fashion, one may prove that, for every p E @, S,(r) + g, is a weak Borel set.

*’

That is, for each nonzero q E @, there exists f E Sj such that f ( ~ ) # 0. Regarding the terminology used in the statement and proof of Lemmas 4.2.16 and 4.2.17, see Appendix I, 51.3. 28

250

Iv.

LINEAR TOPOLOGICAL SPACES

Now, let W be any open set in @. Then, since @ is separable, there exists a sequence {p),} C @, a sequence of positive numbers {Y,}, and a sequence of natural numbers (m,} such that m

w = u {vn +

&&l)).

?l=l

Hence, W is a weak Borel set. Since, in particular, @ itself is a weak Borel set, it follows that every closed set @ - W is also a weak Borel set. 3 Thus, under the hypotheses of Theorem 4.2.16, the totality of weak Borel sets in @ is just the a-algebra generated by the totality of open sets in @.

Lemma 4.2.17. Let @ be a separable countably normed space, with the norm sequence (11 p) \In}. Then, S-,(Y) = {f \lfI\-, r} is a weak Borel set in @+. PROOF. Choose a sequence { P ) ~which } is dense in S, , and consider the set

1

such that limk+mvm, = p) (in the topology of @). Consequently,

that is, f E S-,(Y), Thus, S J r ) = B is a weak Borel set. ] I n particular, we see that Qbnt= S-,(m) is a weak Borel set.

u:=,

50 The Existence of Quasi-Invariant Measures

Now, we shall show how inequality (4.2.12) may be used to derive necessary conditions for the existence of quasi-invariant measures. Here, we only consider Hilbert spaces and number sequence spaces, however, the methods used can, in fact, be applied in an even more general context. Moreover, all the necessary conditions given here are actually sufficient conditions, as will be shown in 95.3.

Theorem 4.2.18. Let G be a separable Hilbert space, with the inner product ( x , y ) , and let b be the totality of weak Borel sets in G. Let 8 be a linear subspace of G, and suppose that 6 itself is a complete


25 1

countably inner product space29 with respect to the sequence of inner products (cp, +), , 7t = 1, 2,..., where (cp, y)l (9,9)2 . Also, suppose that the inclusion mapping T from 6 into G is c o n t i n u o u ~ . ~ ~ For each n, let 8 , denote the completion of 6 with respect to the inner product (cp, +), . Then, the existence of a 6-quasi-invariant finite measure p on (G, 23) implies that, for some natural number n, the mapping T can be extended to a Hilbert-Schmidt type operator from 8 , to G. PROOF. We may assume that p satisfies the condition

0. Restricting 8' and p to Qmt, we obtain a @-quasi-invariant measure space (Qmt, Bmt,pm). Now, in Theorem 4.2.18, we take Q for 8 and Qmt for G, and the desired conclusion follows at once.32 ] Corollary 4.2.20. Let G and 6 be separable Hilbert spaces, where 6 is a linear subspace of G such that the inclusion map T of 6 into G is continuous. Let b be the totality of weak Borel sets in G. If there exists a 8-quasi-invariant finite measure p, on (G, b), then T is a Hilbert-Schmidt operator. PROOF. This corollary is just a special case of Theorem 4.2.18. ] Example 4.2.2, Let I < p < 00, let {a,} be a sequence of positive numbers, and let P({u,}) denote the totality of real number sequences g = (En} which satisfy the condition (4.2.18)

P({U%}) forms a Banach space with respect to the usual coordinatewise linear operations and the norm (4.2.18). In particular, we write ZP in place of P((1)). Let 23 be the a-algebra in P({u,}) generated by the totality of Borel cylinders

(where B represents an arbitrary finite-dimensional Borel set). If s2 Translator's note: The definition of rigged Hilbert space given in Appendix I does not include the completeness of @, which is required here.


253

0, I 5 I = (C,“=II 4,

We may suppose that V = {5 1 so that (4.2.12“) becomes

I

-. d8 J,,,,,,,

Iq)”*,

I En I ME)*

Using the Holder inequality, we deduce that

Multiplying both sides of the above inequality by a , , and summing with respect to n, we get

x (d ) = exp(ih( t))d k ) ,

d E e,

'p E

e7cz(fZ)-

Then, we say that (0, 6) is a dual of (Q, 0) (or that 0 is a dual measure space of Q), and that F is an associated L2-Fourier transform [from L ~ ( Qonto ) !i?k2(0)]. s4.3. Continuous Positive Definite Functions on Linear Topological Spaces

I n this section, we shall, as before, consider only real linear spaces, and shall regard these spaces, in particular, as additive groups. Since any finite-dimensional To linear topological space is topologically isomorphic to a Euclidean space of the same dimension, the Bochner

4.3. Continuous Positive DeJnite Functions

255

theorem provides the general form for continuous positive definite functions on finite-dimensional linear topological spaces. However, for infinite-dimensional linear topological spaces, there still exists no comprehensive theorem concerning the representation of continuous positive definite functions. Nevertheless, there are a number of important results along this line. First, we shall consider the situation for linear spaces in general.

lo Continuous Positive Definite Functions on Linear Spaces. Cylinder Measures T h e usual notion of measure is somewhat too strong for the formulation of certain preliminary results required in the following discussion. Accordingly, we begin by introducing a weaker concept, that is, cylinder measure.

Definition 4.3.1. Let Q be a linear space, let 9 be a linear space consisting of certain linear functionals34 on 8, and let S be the algebra of all Borel cylinders in 6. Suppose that P is a set function on S having the following property: if @ is any finite-dimensional linear subspace of 8, and S ( 0 ) is the o-algebra of Borel cylinders corresponding to 0, then the restriction of P to S ( @ )is a probability measure. Then, we call P a cylinder measure35on 5. Clearly, any cylinder measure P also has the following properties: (i) P(z) 0 for all z E S; (ii) P(sj) = 1; (iii) P is finitely additive. I n fact, (i) and (ii) are obvious from the definition, while (iii) may be seen as follows. If z1,..., z, are disjoint sets in S, then it is clear from the definition of a Borel cylinder that there exists a finite-dimensional subspace @ of 8 such that z1,..., x, E s(@), hence, by the additivity of P on S(@),we have

4%+

**.

+ z,)

= P(X1)

+ ... +

P(Zn),

that is, property (iii). However, P is not, in general, countably additive. But if it happens that P is countably additive, then, using a well-known technique (see, e.g., Halmos [l]), we can extend P to a probability measure (which we also denote by P)on the c~-algebra8 generated by S. Now, let P be any cylinder measure on 5. Let U be a complex-valued function on sj, and suppose there exists a finite-dimensional linear subspace 0 of Q such that U is integrable on the probability measure 34

36

With the usual linear operations. Notice that this is, in fact, a special case of the cylinder measure concept introduced

in $1.3.

256

Iv.


space (fi, S ( @ ) ,P).36 We then say that U is integrable with respect to the S(@),P) is cylinder measure P. I n this case, the integral of U over (8, clearly independent of the choice of @, and will be called the integral of U with respect to the cylinder measure P ; we denote it by

In particular, when P is countably additive, this integral coincides with the ordinary integral of U over the measure space (fi, 8, P).

Definition 4.3.2. Let 6 be a linear space and let f be a complexvalued function on 6 such that, for every finite set ‘pl ,..., y n E 6, the valuef(t,v, t,v,) is a continuous function of the n real variables t , ,..., t , , that is, f is continuous on every finite-dimensional linear subspace3’ of 6. We then say that f is pseudocontinuous on 6.

+ +

Lemma 4.3.1. Let 6 be a linear space, let sj be a linear space consisting of certain real linear functionals on 6, and let P be a cylinder measure on fi. Then, the integral h (4.3.1)

is a pseudocontinuous positive definite function on 6, andf(0) = 1. PROOF. P ( 8 ) = 1, hencef(0) = 1. Furthermore, since

f is positive definite. Finally, given any

v1 ,...,vn E 6, we observe that (4.3.2)

hence, applying the Lebesgue dominated convergence theorem to the probability measure space (9, S(vl ’..., v,), P), we see that (4.3.2) is a continuous function of the real variables tl ,..., t , . 3 If fi contains “sufficiently many” functionals, then the converse of Lemma 4.3.1 also holds, that is, we have the following result. Here, P means the restriction of P to S ( 0 ) . Normally, it is always assumed that finite-dimensional linear spaces are given the Euclidean topology. s7

4.3. Continuous Positive DeJinite Functions

257

Lemma 4.3.2. Let 6 be a linear space and let $3 be a linear space consisting of certain real linear functionals on 8, moreover, suppose that $3 distinguishes the points of 6. Then, given any pseudocontinuous positive definite function f on 8 [with f ( 0 ) = 11, there exists a unique cylinder measure P on 8 such that (4.3.3)

PROOF. Take any linearly independent finite set of vectors cp, ,..., vn in 6, and consider the valuef(t,y, --. t,cp,), regarded as a function of the real variables t, ,..., t, . Since f is positive definite and pseudocontinuous, and f ( 0 ) = 1, it is easily verified that f (t,cpl .-- tncpn) satisfies the conditions of the Bochner-Khinchin theorem, hence, there exists a unique probability measure on real n-dimensional space R, , such that

+

+

+ +

f(tlvl

+ + t,v,) *.*

=

1 1 . * a

exp i(tlE1

+ +

t,5,)

dQ{o,)(tl

t,).

Consider the correspondence T,,, : t -+ (5(y1),..., e(yn)),5 E $3. Since $3 distinguishes the points of @, the image of Ttqj1 is the entire space R, . Define a set function P{,jl on S({yj}) as follows: if A is any Bore1 set in R, , let %J(%:)4 = Q(m,)(A). (4.3.4) Clearly, ($3, S({cpj}),P(mj)) is a probability measure space. Moreover, a straightforward calculation shows that f ' ( q j ) is independent of the particular choice of the basis y 1 ,..., cpn for the finite-dimensional linear subspace under consideration. We now proceed to prove that the family of probability measure spaces constructed in this manner is consistent, that is, if B E S({#,})n S({wj}), #, ,..., t,hrn , w1 ,..., w k E 6, then P(,,)(B)= %JJP)*

Choose a linearly independent set y1 ,..., yn

E

(4.3.5)

8, such that

"

S({A,>) S ( { 4 ) c WvJ).

We need only prove that P(Wj)(B)=

%,m

(4.3.6)

for we may then deduce by the same reasoning that P(qj)(B)= P(u,l(B),

Iv.

258


thereby obtaining (4.3.5). Obviously, we may assume that y j = &, j = 1,2,..., m,n >, m. Since

J J exp &tl + ... + L E , ) .**

dQ($,}(tl ,***,

tm)

+ .-.+ L2fu f(tlV1 + ... + ~,V, + Opm+l+ + OFn) J ... J exp i(tltl + ... + t,t,) ~ Q I ~ ,..., ~ It ,( ~~ ,~

= f(tllCI1 = =

***

it follows that Q(wpl)({(E1

9**'9

En)

for every Bore1 set D in R ,

%J{tI (t(V1)9...,

I (tl

I...)

tm)

ED))

= 8(@,dD>

, that is,

t(V?n>> E Dl) = P($J{t I (E(lCI1)YV

E(lCIm)) E

D)). (4.3.7)

But every set B in S(@,) is of the form {,$ I (,$(&),..., ~),I!( ED}, hence (4.3.7) is equivalent to (4.3.6).Thus, we have shown that the family of measures {P(@,)} is consistent. Define a set function P on S, as follows: if A E S({vj}),let 4 4 )= P ( O j } ( 4 Using the consistency of the family of measure spaces (sj, S ( { y j } ) ,P,,,)), {yj} C 6, and following the reasoning of $1.3, we see that P if unam-

biguously defined. It is then obvious that P is a cylinder measure. Next, we verify that P satisfies (4.3.3).I f g = 0, then (4.3.3)is obvious. If g # 0, then

Finally, the uniqueness of the cylinder measure in (4.3.3) may be seen as follows. T h e values of P on each S({yj}), v1 ,..., yn E 6, are determined by the measure Q(c,) in (4.3.4),and by the Bochner-Khinchin theorem, Q(w,) is uniquely determined by f. ] 2O

Continuity of Cylinder Measures

I n analogy with Definition 3.2.6, we shall now introduce a notion of continuity for cylinder measures.3* 38 In particular, applicable to probability measures on u-algebras of the type described in Definition 4.2.8.

4.3. Continuous Positive Dejnite Functions

259

Definition 4.3.3. Let (ti be a linear topological space with topology 2.Let $3 be a linear space consisting of certain linear functionals on 8 , and let P be a cylinder measure on 9. Suppose that, given any positive number E, there exists a neighborhood V of zero in 8 such that

I

> 1,tE93))
P(U)- E.

PROOF. Let A E S(@),where @ is an m-dimensional linear subspace of (ti. Choose a basis y1 ,..., ym for @, and let 5 denote the totality of Borel subsets of R, . For any B E 5,let

B

fE8L

= {f I(f(P),l>,...,f(P)m))EB,

and define a set function Q on

5 as follows: B E 5.

Q(B)= P(@,

(4.3.15)

Since (6, S(@),P) is a probability measure space, it is easily verified that (R, , 5,Q) is also a probability measure space. Therefore, if D E 5 and D = A, then there exists an open set V in R, such that Y 3 D and (4.3.16) Q(0) > Q(v> - 6. Thus,

r is a weakly open cylinder,

r3

D

= A, and, by

(4.3.15) and

(4.3.16), we have P ( A ) > P(B) - E .

1

Lemma 4.3.8, Let 6 be a linear space, !ij a linear subspace of (ti", and P a cylinder measure on &. Suppose that, given any positive number E , there exists a weakly compact set C , in $3, such that P(2) < 6

(4.3.17)

for every Borel cylinder !i? which does not intersect C,. Then, P is countably additive. PROOF. We shall first prove that, if 2, ,..., z k ,... is any sequence of Borel cylinders in !ijsuch that 2, = 9, then

u;=l

m

1 P(Z,) 2 1 .

(4.3.18)

k=l

Given any positive number E , we know by Lemma 4.3.7 that, for each K, and there is a weakly open cylinder u k such that u k 3

P(2,) > P(U,) Since

u,"~

u k3

-5 . 2k

u,"~z k = !ij2 C,, it follows from the weak compact-

4.3. Continuous Positive Dejinite Functions

265

C, that uF=lU k 3 C, for some m. Thus, the Borel cylinder u s 1U, does not intersect C,. Therefore, by (4.3.17), we have

ness of $3 -

and so, by the finite additivity of P,

Hence,

Letting E -+ 0, we obtain (4.3.18). Now, let {En} be any disjoint sequence of Borel cylinders in $3, such m that Un=lEn = E is also a Borel cylinder. Let E, = $3 - E; then E, is also a Borel cylinder, and En = 9. By (4.3.18), we have

uL0

Hence, by the finite additivity of P,we get

Again, by the finite additivity of P,

hence,

Letting N -+00, we obtain

Iv.

266


Therefore,

c P(En), W

P(E) =

n-1

that is, P is countably additive. 3 T h e following special case of Lemma 4.3.8 will be used in the ensuing discussion.

Corollary 4.3.9. Let 8 be a separable countably normed space,

conjugate space of 8,and P a cylinder measure on Bt. Let (1 ( denote the minus nth norm in 8+,and let SJR) = (6 [ (1 8 (1R) (see Appendix I). Suppose that, given any positive number E , there exist n and R such that (litthe

0. Then NL') and Nh2)are equivalent, and their Kakutani variance inner product is

Now, Nk' is the product of the measures {N,'k),a Theorem 1.4.4,

=

1, 2,...}. Hence, by

(5.1.1 1)

Since

x (a,

-

(l/a,))2

< co is equivalent to

x (I:.z

-

(l/a:''))'

< a,

v. GAUSSIAN MEASURES

286

n.“=,

it follows from (5.1.11) and condition (5.1.9) that p(N:’), NA2’)> 0. Therefore, by Theorem 1.4.4, N,’ and N2’are equivalent. Moreover,

hence, using (1.4.15) and Theorem 1.1.20, we obtain (5.1 .lo). Conversely, suppose that condition (5.1.9) does not hold, for example, suppose that

then there exists a sequence of indices a = 1, 2, ..., such that

a€%,

which we denote by (5.1.12)

Now, in view of (5.1.11), we have

However, by (5.1.12),

nf,

p(Nil’, NL2’)= 0, that is, Hence, by (5.1.13), we conclude that that N,’ and N2’are mutually singular. The case

may be handled in a similar fashion. J Example 5.1.1. Let (I, b, ,p ) be the linear measure space discussed in Example 4.1.2, with probability densities as follows: 1 fu(t> =

(2Tuu)’f4

exp

(-

1 t2

4

uu > 0,

....

a: = 1) 2)

Then, the totality of quasi-invariant points of (I, 8 , , p ) is just 12({l/ua}).

5.1. Some Properties of Gaussian Measures I n fact, let y

E

I, y

= (yl

,...,y, ,...},

287

and form the measure

Then {x,(w), w E Z}(w = {xl(w), ..., x,(w), ...}) are mutually independent Gaussian variables on (I, 23, ,p,,), with mean values yu and variances o, . Using (5.1.9), we see that a necessary and sufficient condition for the equivalence of py and p is

that is, y

E Z2({l/oa}).Moreover,

if y E Z2({l/ou}),then, by (5.1.10), (5. I. 14)

Now, let (I, B , , P) be a canonical Gaussian measure space, with Gaussian variables x,(w), w E I [xa(w) denotes the ath coordinate of a]. The characteristic function q(f) of (I, BU,P) may be constructed as follows. For each f~ It If(.) = fuxa ,x = {xu) E I], applying (5.1.4), we obtain

) E(g(w))));this is a positive Write c ( f , g ) = E ( ( f ( w ) - E ( f ( o ) ) ) ( g ( w definite bilinear functional on It. Also, write

which defines a linear functional on It. Then, the characteristic function may be rewritten as ?(f) = e-c(f.f)+inff)/2,

Lemma 5.1.3.

f 6 p.

(5.1.15)

Let I,, be the totality of real number sequences

x = (xn) such that only a finite number of the x, are not zero; I, forms

a linear space with respect to the usual linear operations (see Example 4.2.1). Let 1 a < 2, and define a norm on I, by

2n+11*

n t, - t ,

(5.1.26)

Since

we get (5.1.27)

We shall show that x E (J~==,+, U g ~Si( K , n, m).Suppose this were not true; then, x E S(K, m,n) for all n j 1, 0 m < 2", that is,

> + 0

0,

P ( S ( K m,n))

(5.1.31)

- /3(bK2"(*-=)),

ST

where p ( t ) denotes the function (2/7~)'/~ exp(-xX2/2) dx. Similarly, we get P(S(K,0, n)) = /3(bK2"(+-@)). (5.1.31 ') However, when t 2, it is easily seen that p ( t ) 6 ( 2 / 7 ~ )e-t. ~ / ~Hence, by (5.1.25), (5.1.31), and (5.1.31'), it follows that, if K > 2*+"/b, then

n = l m=O

n=l m=O 03

< (2/n)lP 1 2" exp(--bK2"(+-")).(5.1.32) n=l

294

v.

GAUSSIAN MEASURES

Clearly, the right-hand side of (5.1.32) tends to zero as K + 00. But R(I) - Ca(Il)C R ( I ) - CaK(Il),hence, letting K -+ 00 in (5.1.32), we obtain (5.1.24). If x E Ca(Il),LY > 0, then x is uniformly continuous on Il , hence x can be uniquely extended to a continuous function on [0, 11, which we also denote by x, moreover, x E Ct’[O, I]. Identifying Ca(Il) with Cou[O,11 in this manner, it is easily seen that the restriction g(a)of dl to Ca(Il)is just the restriction of b to Cp)[O, I]. Let PCm)denote the restriction of P to Ca(Il)(i.e., to Ct)[O, 11). We thus obtain a Gaussian measure space Sa) = (Ck)[O, I], 3(“), P o ) ) , 0 < a < Q. Moreover, { x ( t ) , t E [0, l]} are clearly random variables on S a ) when ; t E Il , they are Gaussian variables with mathematical expectation and covariance function given by (5.1.22) and (5.1.23), where t , s E Il . We shall now prove that, for any t E [0, 11, x ( t ) is also a Gaussian variable on and that (5.1.22) and (5.1.23) are valid for any t, s E [0, I]. Let to E [0, 11 - Il , and choose a sequence {tn}C Il such that t , -+ to . By (5.1.23), we have

hence {x(tn)}, n = 1, 2,..., is a fundamental sequence in J ~ ~ ( S ( ~ ) ) . Moreover, since the functions x in C$)[O, 13 are continuous, limn+o3x(tn) = x(to), consequently, {x(tn)} converges to x(to) in L 2 ( S a ) ) . Hence, it follows by Lemma 5.1.1 that %(to)is a Gaussian variable, furthermore, E((x(t,) - ~ ( t ~ + ) )0,~ )whence it easily follows that (5.1.22) holds for t = to . Now, let so E [0, 11, and choose {sn} C Il such that s, +so ; then we also have E((x(s,) - ~ ( s ~ ) )---t ~ )0, therefore, since

it follows that (5.1.23) holds for s = so , t = to . If &’ E b, then obviously n Clp’[O, 11 E g(a).Define

Clearly, W, is a probability measure on (Co[O,13, 23) which is concentrated on Ct’[O, 11, and is such that (5.1.22) and (5.1.23) hold. Hence, W, satisfies (5.1.21), and is therefore the required Wiener measure. ]

5.2. Equivalence and Perpendicularity

295

55.2. Equivalence and Perpendicularity of Gaussian Measures

l o The Basic Theorem I n 95.1, we considered the question of equivalence for a rather special class of Gaussian measures. I n the present section, we shall consider a more general problem. Let (Q, b)be a measurable space, and (5, , a E a} a family of measurable functions on (Q, b). Let Pl ,P, be two probability measures on (Q, b) such that {& , cy. E a} is a joint determining set for the two measure spaces (Q, b,P,) = S, , v = 1,2. Moreover, suppose that {&(-), 01 E a} is a Gaussian stochastic process with respect to both of the measure spaces S , and S, . We shall investigate the equivalence and perpendicularity of Pl and P , . Henceforth, we shall write EL(%) =

I

dPk(W),

.(W)

s2

Dk(X) = Ek((X

- Ek(X))'),

k = 1, 2.

Let 9 denote the totality of linear combinations of functions { f a , a E a} and real constant functions on Q; obviously, 9 forms a linear space. Define two nonnegative definite bilinear functionals on 9, as follows: ==

(x, r ) k = E k ( q ) ,

1, 2-

Lemma 5.2.1. If the Gaussian measures Pl and Pz are not mutually singular, then there exist positive numbers c1 and c, such that, for all X E 9 ,

(x,

4 2

and (x, 4

1

<
0 such that (5.2.1) holds; then there is a sequence {x,} C 9 such that (xn , x,),

=

1,

(Xn

7

Xn)1-+

O(n

+

I

a).

0, then, by the Chebyshev inequality'

I

' Let P(w) be a probability measure, X ( w ) a random variable and E ( X ) = J X(w)dP(w); then

v. GAUSSIAN

296

MEASURES

If (xn , x,)~ = 0, then x, = 0 almost everywhere (P,), hence P,(Q - A,) = 0. Thus, we have lim P,(Q - A,) n+ m

=

0.

(5.2.3)

Now, let {n,} be any increasing sequence of positive integers. If lim,+a a& = 1, then, writing n in place of n, ,we have Ez((x, - a,J2)---t 0, whence it follows that x, - a, converges to zero in measure (P,). Let N be an integer such that (x, , x,):I4 < and I I - an2 I < for all n 2 N . Then, for w E A,, that is, I x , 1 (x,, x,):/~, we have [ x, - a, I 2 8 for all n 2 N . But P,({w I x, - a, I i}) + O as n + 00, hence P,(A,) + 0. On the other hand, if akk does not converge to 1 as k + 00, then there is a subsequence {n,'} of {n,} such that lim,+a aik. < 1. Since any linear combination of Gaussian variables is also Gaussian, it follows by (5.1.1) that (for brevity, we write n in place of nk')

a

I

Thus, {n,} always contains a subsequence {n,'} such that P2(Ank,) ---t 0. Therefore, we must have lim P2(A,) = 0. n+m

(5.2.4)

But then it follows from (5.2.3), (5.2.4), and Lemma 1.1.21 that PI and P, are mutually singular, contrary to hypothesis. We conclude that (5.2.1) holds for some c1 > 0. Similarly, (5.2.2) holds for some c, > 0. ] We continue to assume that P, and P, are not mutually singular. By virtue of (5.2.1) and (5.2.2), equality almost everywhere (P,) and equality almost everywhere (P2)are equivalent for functions in 9. If we identify functions in S? which coincide almost everywhere (PI),or, what is the same, coincide almost everywhere ( P J , then 9 becomes an inner product space relative to each of the inner products (x, Y ) k , k = 1, 2. Thus, 9 may be regarded as a linear subspace of each of the Hilbert spaces L2(S,) and L2(S2).Let H , and H , denote the closures of S? in L2(S,) and L2(S,), respectively; by Lemma 5.1.1, the elements of H, are also Gaussian variables on S k , k = I , 2. Suppose that [ E Hz,


297

and let {xn>C 9 be a sequence of functions converging to f in the norm . Since { x n ) is a Cauchy sequence relative to (x, x), it follows by (5.2.2) that it is also a Cauchy sequence relative to (x, x ) ,~and hence converges in the norm (x, x ) ~to an element of Hl , which we call T f . It is easily seen that the correspondence ( + T f defines a one-to-one linear mapping T from H2 onto Hl such that the restriction T 1 9 is Using the mapping T , we shall identify the identity mapping on 9. the point-sets of H I and H,, i.e., we shall regard Hl and H , as the Hilbert spaces associated with the respective inner products (x, Y ) and ~ (x, y ) , defined on the same underlying linear space8, which we call H. With this understanding, inequalities (5.2.1) and (5.2.2) are valid for all x E H . (x, x),

)

Lemma 5.2.2. If Pl and P , are not mutually singular, then there exists a positive number K such that, for any finite set of vectors rll )...,rln in H satisfying the conditions

(where

is the Kronecker delta) the following inequality holds, (5.2.5)

PROOF. For convenience, we write El(rjk) = mk , &(Tjk) = “ k . Now, given any set {vI ,..., vn) satisfying the conditions of the lemma, we form the measurable functiong

Actually, for each 5 E H z , one can choose a measurable function x on (a,8)such that x represents f in H , and represents T f in H , . In fact, let {x,J be a sequence of functions in 0 such that x, converges in the norm (., to a function x’ representing 6, and also converges in the norm (., .), to a function x” representing T f . Choosing subsequences, if necessary, we may assume that x, converges almost everywhere (P2) to x’ and also converges almost everywhere ( P , ) to x”. Let x ( w ) = lim x,,(w) if this limit exists, and x,(w) = 0 otherwise. Then the function x has the desired properties. Here, we choose functions representing {vl ,..., 7), in the manner indicated in footnote 8.

v.

29 8

GAUSSIAN MEASURES

on (Q, S).It is easy to calculate thatlo

(5.2.6)

D,(f)

=

C [2 ('k---+4 k

ukz

Write

(5.2.7)

lo For example, D l ( f ) may be calculated as follows: since qk and qk' are mutually independent for k # k', we have


299

On the other hand, by (5.2.1) and (5.2.2), we have

If we write

then it follows from (5.2.6), (5.2.7), and (5.2.10) that there exist positive numbers b, , b, and b (independent of n and q1 ,..., qn) such that

Consequently, by (5.2.8) and (5.2.9), b 1 P”(A,) < r-. b2 m

(5.2.11)

Now, as n and {q, ,..., qn} vary, if the supremum of m were +a,then, for any given positive number E , there would exist some (7, ,..., qn} such that the corresponding number m satisfies m >, (l/b%) max(b, , b,), and so, by (5.2.1 l ) , the corresponding set A, would satisfy P,(A,)

< E,

Pz(Q - A,) < E .

I t would then follow from Lemma 1.1.21 that P , and Pz are mutually singular, contrary to hypothesis. Therefore, we conclude that m has a finite upper bound K . ] Now, let Ho denote the totality of vectors of H which are orthogonal to the vector 1 relative to the inner product (., .), , that is, H , comprises the totality of functions in H having mean value zero relative to the measure Pz . Then Ho is a closed linear subspace of H , , and hence is itself a Hilbert space with respect to the norm (., *), . We form a nonnegative definite bilinear functional on Ho , as follows:

v. GAUSSIAN

300

MEASURES

which shows that B ( ( , v ) is bounded. By a well-known theorem, there exists a bounded self-adjoint operator B on H , such that

4t,7) = (a, 17h for any (, 7 E Ho .

Lemma 5.2.3, If PIand P , are not mutually singular, then B - I (where I is the identity operator in H,) is a Hilbert-Schmidt operator on H,. PROOF. According to Lemma 5.2.2, there exists a positive number K such that -'k)2

= c ( l- B ( r i k , r i r c ) f 2 = c ( ( B - ' I ) 7 k r T / k ) 2 2 k

0, there exists a S > 0 such that If(y) - f ( O ) I < E whenever q E {v I (( T*T/6)v, y ) < I}. Since T*T/S is also a positive nuclear operator, this shows that f l 6 is continuous relative to the 6-topology (see 94.3). Hence, by Theorem 4.3.12, there exists a unique finite measure Pt on (6+, 5,)such that (5.3.3) holds. (3) By Corollary 4.2.20, (ii) implies (i). (4) By Theorem 4.3.11, (ii) implies (iii), ( 5 ) We now proceed to deduce (i) from (iii). Assume that (iii) holds, and consider the continuous function f ( h ) = exp(- &(A, h)l),

h E G.

It is easily verified that this function is positive definite (in fact, it is the characteristic function of a certain Gaussian measure space). Hence, by condition (iii), there exists a unique measure Pt on (6+, 8') such that f satisfies (5.3.3). Next, replace h in (5.3.3) by C:=l t,h, , h, ,..., h, E 6,and define an n-dimensional Bore1 measure P by the equation P(E) = W

t I (t(hl),...> 6(h,)) E El).

We then have

which shows that P is the probability distribution of a Gaussian variable. By (5.1.3),

j,, mJ2dP+(t)= j

Xk2

dP(x) = (h, ,h,)l ,

=

1, 2,...,n,

and if the vectors {hk} are orthogonal relative to the inner product then, by (5.1.6), we get

J,,

(6(h!J2- (A,

!

hR)1)(t(hd2- (A, > Wl) W k,Z= 1,2,...)71,

5 ) = 2(h, > h,)12 &J

where Sk,lis the Kronecker delta. If the operator T is not of Hilbert-Schmidt

(e,

-),

,

9

(5.3.5) type, then, by

v.

318

GAUSSIAN MEASURES

Lemma 11.1.1, T*T is not nuclear. Hence, by Lemma 11.1.6, given any positive integer K, there exist vectors h, ,..., h, E 8, satisfying and (h, , h),= 6,,,(hk h,)l such that (hk 9 h,) = a,,

c

Y

9

hk)l

> K.

n

=

(hk

9

(5.3.6)

k=l

Let Q(t) =

xtzl~ ( I z , ) ~Then, . using (5.3.5), we calculate that

moreover, (h, , hk), < (1 T (I2, hence, by the Chebyshev inequality,

(5.3.7) However, since the (hk} are orthonormal relative to inequality implies

(a,

(5, E ) 2 Q(0.

-), the Bessel (5.3.8)

From (5.3.7) and (5.3.8), we get pt({5NE, 5) 2 - 2 II TI1 d/h))

2 f'+({5 I Q(4) 2

- 2 )I T I1

3 pt({5I I Q(5) - I

4x1)

< 2 II TII a}3)4.

(5.3.9)

On the other hand, m

n {5 I (5, 5 ) 3 K - 2 II T II dK}

= empty set,

K=1

hence lim Pt({5 I

K-tm

(6, 6) 2 K

- 2 11

T 11 dE}) = 0.

(5.3.10)

But by (5.3.6), X > K , therefore (5.3.9) and (5.3.10) are contradictory. Hence (i) holds. ] We shall now establish the converse of Corollary 4.3.14. Corollary 5.3.6.

Let 8 be a real separable countably Hilbert space,

5.3. Gaussian Measures on Linear Spaces

319

5'

the totality of weak Borel sets in (tit. Suppose that, for any positive definite continuous function f on 6 [with f ( 0 ) = I] there exists a probability measure Pt on ((tit, 5') such that (5.3.3) holds for all h E 6. Then (5 is a nuclear space. PROOF.Let (11 * \Im} be the given sequence of norms on 6, and let (tim be the completion of (5 relative to 11 * Ilm. Let m be an arbitrary natural number, and consider the function

Now, suppose that, for every n > m, the imbedding Tmn: (ti, -+ (5, is not of Hilbert-Schmidt type. Then, using part (5) of the proof of Theorem 5.3.5 (replacing (ti by 6, and G by we obtain

I

p+({t II t 11-n =

that is, for all n

00))

=

1~pt({tI II t IIL>, K - 2 II Tmn II dKi>2

> m, Pt(6+-(ti,+) 2

*.

n= :+ ,,

But this contradicts the fact that (St - Ont) is the empty set. We conclude that, for some n > m, the operator Tmnis of HilbertSchmidt type. Therefore, 6 is a nuclear space. ] Next, for number sequence spaces, we establish a partial result analogous to Theorem 5.3.5.

Example 5.3.1. Under the hypotheses of Example 4.2.2, if we 2, then the following two statements are also assume that 1 q equivalent. (i) C a n < 00. (ii) There exists a finite measure on (P((an}),123) which is quasiinvariant with respect to P. Furthermore, let Zqt denote the conjugate space of P, and 5' the totality of weak Borel sets in Zqt. Then, if either (i) or (ii) holds, we have the following conclusion. (iii) For each positive definite continuous function f on P((a,)), there is a unique finite measure Pt on (Int, St) such that, when h E P,

<
. If the characteristic function of a measure is of the type described by Lemma 6.3.4, then this measure is said to be rotationally invariant. I t can be seen from (6.3.20) that any rotationally invariant measure is a superposition of Gaussian measures.22

Theorem 6.3.5. Let R be a complex infinite-dimensional Hilbert space, { W ( z )1 z E A} a conventional free-field system, and 7 any vacuum state vector for this system. Then there exists a finite measure m on co) such that the characteristic functional of {W(z)I z E R) corresponding to 7 is

[t,

(6.3.24)

PROOF. Since r(U)7 = 7, the characteristic functional +(z) has the following property: for U E U, z E R,

Now, if 11 z 11 = 11 zf 1 , there must be some U E U such that z' = Uz, hence, by (6.3.25), I,!J(z) depends only upon 11 z 11. We choose any real linear subspace $j of R such that $jis a real inner product space and R is the complexification of 9. If z is restricted to 5, the function #(z) stiI1 depends only upon 11 z 11. Since 9 is also infinite-dimensional, we know that there is a finite measure m on [0, CQ) such that (6.3.20) holds for all 5 E 9. But since +((), on the entire space R, depends only upon 11 5 11, *l Translator's note: Actually, it suffices to show that the convergence of + , , ( z / ; T p ~ ) is uniform with respect to 7 . See Umemura [ 13 for further details.

380

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

it follows that (6.3.20) holds for all f E R. Using the condition in Lemma 6.2.10 and carrying out a rather intricate calculation, it can be proved that23

M O , t))= 0, whence we obtain (6.3.24). ]

APPENDIX

BACKGROUND MATERIAL ON TOPOLOGICAL

GROUPS AND LINEAR TOPOLOGICAL SPACES

51.1. Pseudometrics, Convex Functions, and Pseudonorms

Definition 1.1.1. Let R be a set, and let p ( x , y ) , x, y E R , be a realvalued function on R x R, satisfying the following conditions: > p(x, y ) 2 0, p(x,). = 0; ( 4 p(x, r) p(x, 4 p (y, 4. Then (i) p is called a pseudometric on R . If R is also a group, and p(y, x ) = p(x-ly, e ) for all x, y E R, then p is said to be a left-invariant pseudometric on R . Right-invariance is similarly defined. Let lpo,,(Y E 'Ir} be a family1 of pseudometrics on R. T h e topology generated by the totality of sets of the form

+

0

Naturally, this subsumes the case in which 'u contains only a single index. This case will not be singled out for special attention in the ensuing discussion.

381

382

APPENDIX

I.

TOPOLOGICAL GROUPS A N D LINEAR SPACES

is called the topology induced by (p, , a E a}.The set R, provided with this topology, is said to be a pseudometric space. If R is a group, and p is a (left and right) invariant pseudometric on R, then R, given the topology induced by p , becomes a topological group.

Definition 1.1.2. Let G be a group, and let N ( g ) , g E G, be a real-valued function on G, satisfying the following conditions: (i) 03 > N ( g ) 2 0, N ( e ) = 0 ; (ii) N ( y x - l ) N ( x ) N ( y ) . Then N is called a convex function (or, in some literature, a pseudonorm) on G. It is easily verified that any convex function also satisfies the condition: (iii) N ( x ) = N ( x - l ) . Given a convex function N on a group G, we introduce the function

-a.Obviously, we may assume that I ( f , U ) < 03, and hence that f ( x > < Z(f9 U )

+c

(1.2.1)

for some point x E U. On the other hand, Z(f, U ) < Z(x,f) n, we have

Hence, the sequence (vl}is fundamental with respect to 11 * 1, for every n, that is, {vl} is a fundamental sequence in @. Therefore, there exists a $ E @ such that

:;\

I1

*

- 'pz Iln = 0,

from which it follows that # = v, and hence v E @. Thus, (1.3.8) holds. Conversely, assume that (1.3.8) holds. If {vm} is a fundamental sequence in @, then {y,J is also a fundamental sequence in @,, (with respect to 11 * ) ,1 for every n. By the completeness of @", there exists rp(") E @" such that (1.3.9) lim 11 'prn - qdn) \In = 0. m+ w

But

@n+l

C Qn , hence q+mf1)

(1 , p + U

- P(")IIn

E

@"

, moreover,

< II ~m - v'~+')Iln + II ~ r -n v ' ~ ) < II ~ r -n v'~+')IIn+l + II v m - Iln (In

v(")

+

0.

Therefore, rp(") = v("+l),n = 1, 2,,.., so that all the elements y(") are @", and (1.3.8) identical and may be denoted by v. Since q E holds, we have v E @. Moreover, by (1.3.9), the sequence {vm} converges to v in @. Thus, @ is complete. ] n, then Qn, is a subspace of @". Thus, regarding anand Qrn If m as Banach spaces with norms (In and 11 * /Irn, respectively, we may define a linear operator Trim, from Qrn to o n ,by

n,"=,

-

Trim : 'p -+

E

'p,

>

Qrn.

It follows from (1.3.2) that T,m (m n) is bounded; we call it the imbedding operator from Gm to @, . A complete countably normed space will be called a countably Banach

1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces

39 1

space. If all the norms 11 * [ I n which define a countably normed space @ are induced by inner products (9, + ) n , that is,

then we call @ a countably inner product space. A complete countably inner product space is called a countably Hilbert space.

Example 1.3.1. Let A denote the finite closed interval [a, b] on the real line. Let K ( d ) denote the totality of infinitely differentiable complex-valued functions y ( x ) on the real line which vanish outside of d. Obviously, K ( d ) forms a linear space with respect to the ordinary linear operations. For any rp, E K ( d ) , we define

+

(q,

=

v=o

I

pl'Y'(t)

p ( t ) dt,

fl

= 1, 2

,...,

(I.3.10)

A

where yt0) means just rp itself. We write 1) y \ I n = [(rp, rp)n]1/2. Clearly, K ( d ) , with the sequence of norms (11 n = 1, 2, ...}, forms a countably inner product space. Consider those functions rp, defined on the real line, which vanish outside of A , have an absolutely continuous derivative of order n - 1, and such that rpcn) € L 2 ( d ) .Denote the totality of such functions p by &(A). Define an inner product on K n ( d )by formula (1.3.10). Clearly, &(A), with the ordinary linear operations and the inner product (1.3.10), forms a Hilbert space. In fact, Kn(d)is just the completion of K(A) with respect to the norm 11 ]In . Moreover, K,(d) 3 &(A) 3 ... 3 &(A) 3 -.* and

-

-

m

W) =

Kn(4 n=l

consequently, K ( d ) is a countably Hilbert space. 2O

Spaces of Continuous Linear Functionals

Let @ be a countably normed space, with norm sequence (11 * 1 I n , n = 1,2, ...I, and, for every a, let @% be the completion of @ with respect to the norm 11 * ) I n . Then On is, of course, a Banach space; we denote its conjugate space by Qnt, and the norm in Qnt by

If rn 2 n, any element of Qnt, when restricted to Qrn,is also continuous

392

APPENDIX

I.

TOPOLOGICAL GROUPS AND LINEAR SPACES

with respect to the norm 11 * [Irn, and similarly, when restricted to @, is continuous with respect to the topology of @. Thus, denoting by Gt the linear space consisting of all continuous linear functionals on @, we have

moreover, we assert that

(I.3.12) In fact, if F E Qt, then there is a positive number E such that p E @, 0) < E implies I F(p)l < 1. Choose no sufficiently large, so that Zz=n,+l(1/2m)< ~ / 2 and , choose a positive number 6 < 1, such that a/( 1 6) < c/2. Then, when I( p llnO < 6, we have p(p,

+

$ E @,

(I.3.13)

where M = 1/a2. Forming the completion of @ with respect to the new inner product (p, $), we obtain a Hilbert space H 3 @. We may, in the usual manner, identify H with its own conjugate space Ht. The restriction to Qi of any element ( of H t (= H) is a linear functional on @, defined explicitly by p -+ (9,5). Moreover, by (1.3.13),

hence

8 : p -+(9,0,p E @,

is a continuous linear functional on @,

1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces in fact, 5 E ,@; of spaces

. Thus, we obtain

393

an imbedding of H in Qt. The triple

@CHC@+

(1.3.14)

is called a rigged Hilbert space.

Example 1.3.2. Define an inner product on K(A) by J A

T h e completion of K(A) with respect to (rp, #) is just La(A),and K(A)t is the space of generalized functions on A. Thus, we obtain the rigged Hilbert space K(A) C L 2 ( A )C Kt(d). m

Let K = (Jn=l K(A,), A , = [-n, n]. We specify that a sequence {rp,} converges to rp in K, provided that, for some K, {rp,} C K(d,), rp E K(A,), and that {rp,} converges to 'p in the topology of K(A,) (as defined in Example 1.3.1). Accordingly, we say that a functional f on K is continuous provided that f(rpn) +f(rp) whenever {rp,} converges to rp in the above sense. I t can be proved that K is the exact union of the sequence of spaces {K(d,),n = 1,2,...} (see Definition 4.3.5).

Example 1.3.3.

APPENDIX

BACKGROUND MATERIAL ON FUNCTIONAL ANALYSIS IN HILBERT SPACES

§11.1. Operators of Hilbert-Schmidt Type, Nuclear Operators, and Equivalence Operators

lo Basic Properties of Hilbert-Schmidt Operators and Nuclear Operators I n this subsection, we restrict our considerations to Hilbert spaces. Let H and G be two Hilbert spaces, T a completely continuous linear operator from H to G. Let T* denote the adjoint of T. Then T*T is a completely continuous self-adjoint linear operator from H to H, moreover, for any cp E H,

that is, T*T is a positive operator. According to the spectral resolution theorem for completely continuous self-adjoint operators, there is an 394

11.1. Hilbert-Schmidt-Type, Nuclear, and Equivalence Operators

395

orthonormal system of eigenvectors {en) of T*T, with corresponding eigenvalues An2 > 0, such that, for any 'p E H, (11.1.1)

Let g , = (l/A,) Ten . Then, 1

( g m 9 gn) = -(Ten U r n

, Te,)

1 XnXm

= -( T * T e n

, em)

=

,

is the Kronecker delta, that is, {g,} is an orthonormal system where in G. Now, for any 'p E H, there is a vector u 1{en)such that

Since u

1{en),u I T*Tu, that is, (Tu,Tu) = 0. Hence,

(11.1.2)

where A,

> 0 and limn+mA,

= 0.

Definition 11.1 .I. If, in (11.1.l), we have C,"cl An2 < co, then the operator T is said to be of Hilbert-Schmidt type (written briefly as H-S type); if Z:-l A, < co,then T is said to be a nuclear operator. Obviously, every nuclear operator is of H-S type, but an operator of H-S type is not necessarily nuclear. Any continuous linear operator of finite rank (i.e., having a finite-dimensional range) is, of course, nuclear. Lemma 11.1.1. Let T be a bounded linear operator from a Hilbert space H to a Hilbert space G. Then T is of H-S type if and only if T * T is nuclear. PROOF. The necessity of the condition is obvious. Conversely, if T * T is nuclear, then there exist numbers A, > 0, C An2 < 00, and an orthonormal system {en} in H , such that (11.1.1) holds. Hence, (11.1.2) also holds. It then follows that T is completely continuous and is of H-S type. ] Lemma 11.1.2. Let T be a completely continuous linear operator from a Hilbert space H to a Hilbert space G. Then, T is of H-S type if

396

APPENDIX

11.

FUNCTIONAL ANALYSIS I N HILBERT SPACES

and only if there exists a positive number 1M such that, for any orthonormal system {p),} in H , (11.1.3)

PROOF. Assume that T is of H-S type. Then, there exist orthonormal systems {e,} and {gy} in H and G, respectively, such that (11.1.2) holds. By (11.1.2) we have, for any p) E H , (11.1.4)

Let (cppz} be an arbitrary orthonormal system in H. Then, by the Bessel inequality,

C

I(vn >

ev)12

< It eu t12

(11.1.5)

= 1.

n

Using (11.1.4) and (11.1.5), we get

c II

Tvn

(I2

C

=

n

v

n

I(vn

ev)12

B*gn)

d MI Cen 11'

+ II B*gn 112)*

By Lemma 11.1.3, B* is of H-S type, hence, by Lemma 11.1.2, we have

1 < 31 (T II Cen + 1II B*gn An

112

112)

< 00,

as required. 3 Remark. It can also be proved that any nuclear operator can be resolved into the product of two H-S type operators. Next, we give another useful sufficient condition for an operator to be nuclear.

Theorem 11.1.8. Let T be a linear operator from a Hilbert space H to a Hilbert space G. If there exist families of vectors {h,} in H and {#,} in G, such that T has the representation

then T is a nuclear operator. PROOF. In view of (11.1.13), it may be assumed that {h,) and {#,} are countable. Form the operator

Then T, is a continuous linear operator of finite rank, and is therefore completely continuous. Moreover, since

1) c m

II(T- Tn)v 11

=

u=n+l

1 u*12+1 c 11

(v, h v ) +u
-1,

where {en) is an orthonormal system in H . Then, one easily calculates that ((A*A)1/2- I)p, =

But (I

C((1 + An)1/2

- l)(v, en) en

.

+ hn)1l2 - 1 is a real number, moreover C ((1 + hn)1/2-

0) of X . Consequently, X is the sum of I matrices of rank one, that is,

.....I ,

(~~~.Y(~”)””~zz,.v~~v~l’*)),,.~~l.a

” = 1, 2,*..,1.

406

11. FUNCTIONAL

APPENDIX

ANALYSIS IN HILBERT SPACES

From (11.2.5), we deduce at once that the Schwarz inequality

w’z

I(@,Y9l d ((a,@)(K

n:El

is valid. Consequently, if @ E @ sj, , (@, @) = 0, then (@, for every Y E @ $5,. I n particular, choosing Y = f“E s j v 9 we get

n:E,

,

fI O f u )= 0,

= (@,

@(fl ,.*.,fn)

Y)= 0

nl10 fu

v=l

hence @ = 0. Thus, we have proved that (@, Y) is an inner product. ] Throughout the following, @ 6, will be regarded as an inner product space with respect to the inner product (11.2.3), and we shall write

ni:l

I/ @ 11

=

(@,

a)l’a*

It is easily verified that

Moreover, since

for any @ E

nLEl @ sj, , it follows that

Lemma 11.2.2, in

Let {GS},s = 1, 2,... be a fundamental sequence

niEl @ sj, . Then, there exists a functionaI @ nr=, E

,...,fn)

@(f1

= lim r+m @,(fl

for every n-tuple of vectorsf, E 43, , v PROOF. By (11.2.6), we have

=

,...,f n )

I , 2,..., n.

Consequently, for any fixed n-tuplef, ,...,fn , lims+m@,(f1 ,...,fn) exists; denote this limit by @(fl ,...,fn). It is easily verified that the functional @(fi

nL G3 1 nr-, @ sju be the totality of functionals @ in nr=l

,...,fn) E

Let

sju ‘

407

11.2. Tensor Products of Hilbert Spaces

satisfy the following condition: there exists a fundamental sequence {OS} In in @ sjY such that

nV=,

for every n-tuple f v E sj, , v = 1, 2, ..., n. When this condition is satisfied, we say that {QS} is a defining sequence for @. If Y E @ sj, , and {Ys} is a defining sequence for Y, we define the inner product of @ and Y by

n,”=,

(@, Y)

=

lim (as, Y,).

n-m

(11.2.8)

I t is easily verified that the above limit exists, is independent of the choice of defining sequences for @ and Y, and does, in fact, define an inner product on @ sj,. Moreover, from Lemma 11.2.2, the reader can easily deduce the following result.

n;=,

n,”El

Theorem 11.2.3. @ sj, forms a Hilbert space with respect to the inner product (11.2.8). I n fact, If=, Q sjv is a completion of the inner product space

I-IS Q 43,.

n;=, Q sj, is called the tensor

Definition 11.2.2. T h e Hilbert space product of the inner product spaces sj, ,..., 8,.

Example 11.2.1. Let 52, = (G,, 23, , Pv),v = 1, 2,..., n, be a family be the Hilbert space formed of measure spaces. For every Y, let L2(QV) by the totality of quadratically integrable functions on 52” with the usual linear operations and the inner product

Let 52 = (G, 23, P ) be the product of the measure spaces 52, ,..., 52, . Then, any @ €L2(52)may be regarded as a functional @(fl ,...,f,), f, E L ~ ( Q , v) ,= 1, 2 ,...,n, namely, W

l

,...,fn)

=

I

~@(W)fi(%)

**..fn(Wn)W W ) ,

R

w = (wl

,..., wJ.

n;=,

In accordance with this identification, @ f, is just the function -..f,(w,) in L2(52). It is easily seen that the tensor product @ L2(QV) is, in this case, simply L2(52).

fl(wl)

n:=,

408

APPENDIX

11. FUNCTIONAL

ANALYSIS IN HILBERT SPACES

We now consider tensor products of identical factors $jl = = 5, =8. n In this case, we write &(,) in place of @ 8,; the elements of 5 j ( l E ) are called tensors of ordev n over 5. Let Zn denote the symmetric group of degree n, that is, the group of all permutations of the symbols 1,2, ..., n. If T E Z, , we denote the result of applying the permutation T to the symbols 1,..., n by ~ ( 1,..., ) rr(n), respectively. Now, form a 0 5,, as follows: if linear operator V ( " ) ( T ) in

nv=l

n:El

then

c I

V'"'(7r)aJ=

n

@f3?,.

p=1 v = l

ni!l

It is easily seen that V(,)(T)is a unitary operator from @ 9, onto itself. Since @ &, is dense in $j(,), V n ) ( v can ) be uniquely extended to a unitary operator from & ( l Z ) onto itself. Furthermore, the correspondence rr -+ V(,)(T-~) is a unitary representation of the symmetric group in the Hilbert space (for the definition of a unitary representation, see 511.3). Consider the bounded linear operator

n:cl

in

&(").

Actually, S, is a projection operator. I n fact, since V(,)(T)*=

P ) ( T we - ~have ), 1 s,* = c V(")(,-1) = s, . n! ~

,{TT' I T' E Z},

On the other hand, for any T E Z, Consequently, V'"'(7r)s,

=

I

= {T'T T' E Z },

1 -1 V(n)(7rT')= s, = SnV(n)(7r), nI ,,,

= 2,

.

(11.2.9)

Therefore, since Zn is of order n ! , we obtain s , 2

=

J&z

V'"'(77)s, = s,

I

n

Formula (11.2.9) also shows that every vector of M , = Sn$P), the range of S,, is invariant with respect to all the operators V n ) ( r r ) ,

11.3. Unitary Representations of Groups

409

. The elements of M , are called symmetric tensors of order n, and the projection S, is called the symmetrization operator. M , , being a closed subspace of a Hilbert space $j("), is itself a Hilbert space with respect to the inner product (@, Y).

rr E Z,

Example 11.2.2. If, in Example 11.2.1, we take Q, = Q, = = Q,, then the symmetric tensors of order n are just the symmetric functions f ( w ) in L2(sZ),that is, those which satisfy

for all permutations rr E En .

Lemma 11.2.4. Let {g,} be a complete orthonormal system in the Hilbert space $j. Then,2 '!

)"'S,(gp

(k,!k,! ... kl!

@

... @g!'),

0 < k,,

1

1 k, = n v=l

(11.2.10)

forms a complete orthonormal system in the corresponding space M , of symmetric tensors of order n. PROOF. Clearly, (11.2.10) is an orthonormal system; we need only prove that it is complete. Suppose that g E M , is orthogonal to all of the vectors (11.2.10). Then

But every vector in $(), is a limit of linear combinations of vectors of the form g$ @ *.- @ g t t , moreover, S,g = g. Consequently, g is orthogonal to the whole of a(,), that is, g = 0. This shows that (11.2.10) is complete. ] 511.3. Unitary Representations of Groups

lo General Concepts Definition IL3.1. Let 6 be a group, H a Hilbert space. Suppose that, to every element g E 8 , there corresponds a unitary operator U ( g ) * Here,:g

represents gl 0 ... 0g, kl IBCtOrS

.

410

APPENDIX

11. FUNCTIONAL

ANALYSIS I N HILBERT SPACES

on H , and that, for any g , ,g , E 6, we have U ( g l ) U ( g , ) Then the correspondence

=

U(g,g,).

is called a unitary representation of the group 6 in the Hilbert space H. If we let U denote the totality of unitary operators in H , then a unitary representation U of 6 in H is simply a homomorphism of 6 into U. If there exists no nontrivial closed linear subspace of H which is invariant under all the operators U(g), g E 8,then the representation U is said to be irreducible. Clearly, U is irreducible if and only if { U ( g ) I g E G}' (see $2.3, 2O) consists of just the operators X I , where X is a number and I is the identity operator in H . Now, suppose that 6 is a topological group, and U is a unitary representation of 6 in a Hilbert space H. If, for every fixed pair of vectors T, ht, E H , the function (

wvv, *I

is continuous on 6, then the representation U is said to be weQkly continuous. I n other words, U is a weakly continuous representation provided that, when U is given the weak topology (see $2.3, lo), the correspondence U : 6 -+ U is a continuous homomorphism. If, for every fixed q~ E H , the correspondence h + U(h)v,

hE8

is a continuous mapping from 8 to H (relative to the norm of H ) , then the representation U is said to be strongly continuous. I n other words, the representation U is strongly continuous provided that U is a continuous mapping from 6 to U when the latter is given the strong topology (see $2.3, lo). Obviously, any strongly continuous unitary representation is also weakly continuous. Actually, the converse is also true.

Lemma 11.3.1. A weakly continuous unitary representation is also strongly continuous. PROOF. Let U be a weakly continuous unitary representation of a topological group 6 in a Hilbert space H . Using the definition of a unitary representation, one readily calculates that, for any q E H , go,g E 8,

11.3. Unitary Representations of Groups

41 1

Let E be an arbitrary positive number. Since U is weakly continuous, there exists a neighborhood V of the identity e in 8,such that

for every h e V . But ( U ( e ) y ,y) = ( y , y), hence, if g E ~ , V(which is a neighborhood of go), it follows from (11.3.1) and (11.3.2) that

Therefore, U is strongly continuous. ] Let U , :g -+ U,(g), g E G , k = 1, 2, be unitary representations of a group G in the respective Hilbert spaces H I and H , . If there exists a unitary operator Q from HI onto H , such that U,(g) = QU,(g) Q-I, g E G , then U , and U , are said to be unitarily equivalent. 2 O One-parameter Groups of Unitary Operators

Let H be a Hilbert space, and { U ( t ) , -co unitary operators in H such that U(t1

+ tz)

=

U(t1) U(t,),

-CO

< t < co} a family of

< t , , t z < CO, U(0) = I. (11.3.3)

Then, { U ( t ) , - co < t < a} is called a one-parameter group3 of unitary operators in H . In other words, saying that (U(t ),--a3 < t < a>is a one-parameter group of unitary operators in H means that the correspondence t --t U ( t )is a unitary representation of R (the additive group of real numbers) in H . If this representation is strongly (or weakly) continuous, then we say that the one-parameter group of unitary operators { U ( t ) , -a < t < a}is strongZy (or weakZy) continuous. If, for every fixed pair of vectors y , $ E H , (U(t)T, $1,

--co

Measure and Integration Theory on Infinite Dimensional Spaces

Measure and integration theory on infinite-dimensional spaces

Measure and integration theory on infinite-dimensional spaces: Abstract harmonic analysis

Measure, integration and function spaces

Measure, Integration and Function Spaces

Measure, Integration and Function Spaces

Measure, Integration And Function Spaces