Harmonic Analysis on Semigroups

Christian Berg Jens Peter Reus Christensen Paul Ressel

Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions

Springer-Verlag New York Berlin Heidelberg Tokyo

Christian Berg Jens Peter Reus Christensen Matematisk Institut K0benhavns U niversitet Universitetsparken 5 DK-2100 K0benhavn ~ Denmark

Editorial Board P. R. Halmos Managing Editor Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A.

Paul Ressel Mathematisch-Geographische FakulHit Katholische UniversiHit EichsHitt Residenzplatz 12 D-8078 EichsHitt Federal Republic of Germany

F. W. Gehring

C. C. Moore

Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.

Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A.

AMS Classification (1980) Primary: 43-02, 43A35 Secondary: 20M14, 28C15, 43A05, 44AI0, 44A60, 46A55, 52A07, 60E 15 Library of Congress Cataloging in Publication Data Berg, Christian Harmonic analysis on semigroups. (Graduate texts in mathematics; 100) Bibliography: p. Includes index. 1. Harmonic analysis. 2. Semigroups. I. Christensen, Jens Peter Reus. II. Ressel, Paul. III. Title. IV. Series. QA403.B39 1984 515'.2433 83-20122 With 3 Illustrations.

© 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. Typeset by Composition House Ltd., Salisbury, England. Printed and bound by R. R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.

9 87 6 54 32 1

ISBN 0-387-90925-7 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-90925-7 Springer-Verlag Berlin Heidelberg New York Tokyo

Preface

The Fourier transform and the Laplace transform of a positive measure share, together with its moment sequence, a positive definiteness property which under certain regularity assumptions is characteristic for such expressions. This is formulated in exact terms in the famous theorems of Bochner, Bernstein-Widder and Hamburger. All three theorems can be viewed as special cases of a general theorem about functions qJ on abelian semigroups with involution (S, +, *) which are positive definite in the sense that the matrix (qJ(sj + Sk)) is positive definite for all finite choices of elements Sl' . . . , Sn from S. The three basic results mentioned above correspond to (IR, +, x* = -x), ([0,00[, +, x* = x) and (No, +, n* = n). The purpose of this book is to provide a treatment of these positive definite functions on abelian semigroups with involution. In doing so we also discuss related topics such as negative definite functions, completely monotone functions and Hoeffding-type inequalities. We view these subjects as important ingredients of harmonic analysis on semigroups. It has been our aim, simultaneously, to write a book which can serve as a textbook for an advanced graduate course, because we feel that the notion of positive definiteness is an important and basic notion which occurs in mathematics as often as the notion of a Hilbert space. The already mentioned Laplace and Fourier transformations, as well as the generating functions for integervalued random variables, belong to the most important analytical tools in probability theory and its applications. Only recently it turned out that positive (resp. negative) definite functions allow a probabilistic characterization in terms of so-called Hoeffding-type inequalities. As prerequisites for the reading of this book we assume the reader to be familiar with the fundamental principles of algebra, analysis and probability, including the basic notions from vector spaces, general topology and abstract

vi

Preface

measure theory and integration. On this basis we have included Chapter 1 about locally convex topological vector spaces with the main objective of proving the Hahn-Banach theorem in different versions which will be used later, in particular, in proving the Krein-Milman theorem. We also present a short introduction to the idea of integral representations in compact convex sets, mainly without proofs because the only version of Choquet's theorem which we use later is derived directly from the Krein-Milman theorem. For later use, however, we need an integration theory for measures on Hausdorff spaces, which are not necessarily locally compact. Chapter 2 contains a treatment of Radon measures, which are inner regular with respect to the family of compact sets on which they are assumed finite. The existence of Radon product measures is based on a general theorem about Radon bimeasures on a product of two Hausdorff spaces being induced by a Radon measure on the product space. Topics like the Riesz representation theorem, adapted spaces, and weak and vague convergence of measures are likewise treated. Many results on positive and negative definite functions are not really dependent on the semigroup structure and are, in fact, true for general positive and negative definite matrices and kernels, and such results are placed in Chapter 3. Chapters 4-8 contain the harmonic analysis on semigroups as well as a study of many concrete examples of semigroups. We will not go into detail with the content here but refer to the Contents for a quick survey. Much work is centered around the representation of positive definite functions on an abelian semigroup (S, +, *) with involution as an integral of semicharacters with respect to a positive measure. It should be emphasized that most of the theory is developed without topology on the semigroup S. The reason for this is simply that a satisfactory general representation theorem for continuous positive definite functions on topological semigroups does not seem to be known. There is, of course, the classical theory of harmonic analysis on locally compact abelian groups, but we have decided not to include this in the exposition in order to keep it within reasonable bounds and because it can be found in many books. As described we have tried to make the book essentially self-contained. However, we have broken this principle in a few places in order to obtain special results, but have never done it if the results were essential for later development. Most of the exercises should be easy to solve, a few are more involved and sometimes require consultations in the literature referred to. At the end of each chapter is a section called Notes and Remarks. Our aim has not been to write an encyclopedia but we hope that the historical comments are fair. Within each chapter sections, propositions, lemmas, definitions, etc. are numbered consecutively as 1.1,1.2,1.3, ... in §1, as 2.1,2.2,2.3, ... in §2, and so on. When making a reference to another chapter we always add the number of that chapter, e.g. 3.1.1.

vii

Preface

We have been fascinated by the present subject since our 1976 paper and have lectured on it on various occasions. Research projects in connection with the material presented have been supported by the Danish Natural Science Research Council, die Thyssen Stiftung, den Deutschen Akademischen Austauschdienst, det Danske U ndervisningsministerium, as well as our home universities. Thanks are due to Flemming Topsq,e for his advice on Chapter 2. We had the good fortune to have Bettina Mann type the manuscript and thank her for the superb typing.

March 1984

CHRISTIAN BERG lENS PETER REUS CHRISTENSEN PAUL RESSEL

Contents

CHAPTER 1

Introduction to Locally Convex Topological Vector Spaces and Dual Pairs §1. Locally Convex Vector Spaces §2. Hahn-Banach Theorems §3. Dual Pairs Notes and Remarks

1 1 5 11 15

CHAPTER 2

Radon Measures and Integral Representations §1. §2. §3. §4. §5.

Introduction to Radon Measures on Hausdorff Spaces The Riesz Representation Theorem Weak Convergence of Finite Radon Measures Vague Convergence of Radon Measures on Locally Compact Spaces Introduction to the Theory of Integral Representations Notes and Remarks

16 16 33 45 50 55 61

CHAPTER 3

General Results on Positive and Negative Definite Matrices and Kernels

66

§1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels §2. Relations Between Positive and Negative Definite Kernels §3. Hilbert Space Representation of Positive and Negative Definite Kernels Notes and Remarks

66 73 81 84

Contents

X

CHAPTER 4

Main Results on Positive and Negative Definite Functions on Semigroups §1. Definitions and Simple Properties §2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups §3. Negative Definite Functions on Abelian Semigroups §4. Examples of Positive and Negative Definite Functions §5. r-Positive Functions §6. Completely Monotone and Alternating Functions Notes and Remarks

86 86 92 98 113 123 129 141

CHAPTER 5

Schoenberg-Type Results for Positive and Negative Definite Functions §1. §2. §3. §4. §5.

Schoenberg Triples Norm Dependent Positive Definite Functions on Banach Spaces Functions Operating on Positive Definite Matrices Schoenberg's Theorem for the Complex Hilbert Sphere The Real Infinite Dimensional Hyperbolic Space Notes and Remarks

144 144 151 155 166 173 176

CHAPTER 6

Positive Definite Functions and Moment Functions §1. §2. §3. §4. §5.

Moment Functions The One-Dimensional Moment Problem The Multi-Dimensional Moment Problem The Two-Sided Moment Problem Perfect Semigroups Notes and Remarks

178 178 185 190 198 203 222

CHAPTER 7

Hoeffding's Inequality and Multivariate Majorization §1. The Discrete Case §2. Extension to Nondiscrete Semigroups §3. Completely Negative Definite Functions and Schur-Monotonicity Notes and Remarks

226 226 235 240 250

CHAPTER 8

Positive and Negative Definite Functions on Abelian Semigroups Without Zero §1. Quasibounded Positive and Negative Definite Functions §2. Completely Monotone and Completely Alternating Functions Notes and Remarks

252 252 263 271

References

273

List of Symbols

281

Index

285

CHAPTER 1

Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

§1. Locally Convex Vector Spaces The purpose of this chapter is to provide a quick introduction to some of the basic aspects of the theory of topological vector spaces. Various versions of the Hahn-Banach theorem will be used later in the book and the exposition therefore centers around a fairly detailed treatment of these fundamental results. Other parts of the theory are only sketched, and we suggest that the reader consult one of the many books on the subject for further information, see e.g. Robertson and Robertson (1964), Rudin (1973) and Schaefer (1971). 1.1. We assume that the reader is familiar with the concept of a vector space E over a field IK, which is always either IK = IR or IK = C, and of a topology (!) on a set X, where (!) means the system of open subsets of X. Generally speaking, whenever a set is equipped with both an algebraic and a topological structure, we will require that the structures match in the sense that the algebraic operations become continuous mappings. To be precise, a vector space E equipped with a topology (!) is called a topological vector space if the mappings (x, y) ~ x + y of E x E into E and (A, x) ~ AX of IK x E into E are continuous. Here it is tacitly assumed that E x E and IK x E are equipped with the product topology and IK = IR or IK = C with its usual topology. A topological vector space E is, in particular, a topological group in the sense that the mappings (x, y) ~ x + y of E x E into E and x ~ - x of E into E are continuous. For each u E E the translation L u : X ~ X + u is a homeomorphism of E, so if fJl is a base for the filter 0/1 of neighbourhoods of zero, then u + fJl is a base for the filter of neighbourhoods of u. Therefore the whole topological structure of E is determined by a base of neighbourhoods of the origin.

2

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

A subset A of a vector space E is called absorbing if for each x E E there exists some M > 0 such that x E AA for all A E lK with IAI ~ M; and it is called balanced, if AA f; A for all A E lK with I AI ~ 1. Finally, A is called absolutely convex, if it is convex and balanced.

1.2. Proposition. Let E be a topological vector space and let 0/1 be the filter of neighbourhoods of zero. Then: (i) every V E 0/1 is absorbing; (ii) for every V E 0/1 there exists V E 0/1 with V + V f; U; (iii) for every V E 0/1, b(V) = nlJlI~ 1 J1V is a balanced neighbourhood of zero

contained in V.

°

PROOF. For a E E the mapping A ~ Aa of lK into E is continuous at A = and this implies (i). Similarly the continuity at (0,0) of the mapping (x, y) ~ x + y implies (ii). Finally, by the continuity of the mapping (A, x) ~ AX at (0, 0) E lK x E we can associate with a given U E 0/1 a number e > and V E 0/1 such that AV f; U for I AI ~ c. Therefore

°

eV

f;

b(V)

~

V

so V contains the balanced set b(V) which is a neighbourhood of zero because e V is so, x ~ ex being a homeomorphism of E. D From Proposition 1.2 it follows that in every topological vector space the filter 0/1 has a base of balanced neighbourhoods. A topological vector space need not have a base for 0/1 consisting of convex sets, but the spaces we will discuss always have such a base.

1.3. Definition. A topological vector space E over lK is called locally convex if the filter of neighbourhoods of zero has a base of convex neighbourhoods. 1.4. Proposition. In a locally convex topological vector space E the filter of neighbourhoods of zero has a base gj with the following properties: (i) Every V E gj is absorbing and absolutely convex. (ii) If V E gj and A 9= 0, then AV E (JI.

Conversely, given a base (JI for a filter on E with the properties (i) and (ii), there is a unique topology on E such that E is a (locally convex) topological vector space with gj as a base for the filter of neighbourhoods of zero. PROOF. If V is a convex neighbourhood of zero then b(V) is absolutely convex. If gjo is a base of convex neighbourhoods, then the family gj = {Ab( V) I V E gj 0' A 9= o} is a base satisfying (i) and (ii). Conversely, suppose that gj is a base for a filter ff on E and satisfies (i) and (ii). Then every set V E ff contains zero. The only possible topology on E which makes E to a topological vector space, and which has ff as the filter of neighbourhoods of zero, has the filter a + ff as filter of neigh-

3

§l. Locally Convex Vector Spaces

bourhoods of a E E. Calling a nonempty subset G s; E "open" if for every a E G there exists U E gj such that a + U S; G., it is easy to see that these "open" sets form a topology with a + f7 as the filter of neighbourhoods of a., and that E is a topological vector space. D In applications of the theory of locally convex vector spaces the topology on a given vector space E is often defined by a family of seminorms.

1.5. Definition. A function p: E following properties:

~

[0,

00 [

is called a seminorm if it has the

(i) homogeneity:p(:1x) = 1:1lp(x)for:1E lK,XEE; (ii) subadditivity: p(x + y) ~ p(x) + p(y) for x, y E E. If, in addition, P- 1( {O}) = {O}, then P is called a norm. If P is a seminorm and (X > 0 then the sets {x EEl p(x) < (X} are absolutely convex and absorbing. For a nonempty set A S; E, we define a mapping PA: E ~ [0, 00] by

PA(X) = inf{:1 > Olx E :1A} (where PA(X) = 00, if the set in question is empty). The following lemma is easy to prove. 1.6. Lemma. If A

S;

E is

(i) absorbing, then PA(X) < 00 for x E E; (ii) convex, then PAis subadditive; (iii) balanced, then PAis homogeneous, and

{x E ElpA(X) < I}

S;

A

S;

{x E ElpA(X)

~

I}.

If A satisfies (i)-(iii) then PA is called the seminorm determined by A. A seminorm P satisfies Ip(x) - p(y) I ~ p(x - y). In particular, if E is a topological vector space then P is continuous if and only if it is continuous at 0 and this is equivalent with {xlp(x) < (X} being a neighbourhood of zero for one (and hence for all) (X > o. We will now see how a family (Pi)iel of seminorms on a vector space E induces a topology on E.

1.7. Proposition. There exists a coarsest topology on E with the properties that E is a topological vector space and each Pi is continuous. Under this topology E is locally convex and the family of sets

is a base for the filter of neighbourhoods of zero.

4

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

PROOF. Let gj denote the above family of sets. Then gj is a base for a filter on E having the properties (i) and (ii) of Proposition 1.4, and the unique topology asserted there is the coarsest topology on E making E to a topological vector space in which each Pi is continuous. D The above topology is called the topology induced by the family (pJiEI of seminorms. Note that in this topology a net (x a ) from E converges to x if and only if lima plx - x a ) = 0 for all i E I. The topology of an arbitrary locally convex topological vector space E is always induced by a family of seminorms, e.g. by the family of all continuous seminorms as is easily seen by 1.4 and 1.6. 1.8. Proposition. Let E be a locally convex topological vector space, where the topology is induced by a family (pJiEI of seminorms. Then E is a Hausdorff space if and only iffor every x E E\ {OJ there exists i E I such that plx) =f O. PROOF. Suppose x =f y and that (pJiEI has the above separation property. Then there exist i E I and G > 0 such that Pi(X - y) = 2c;. The sets

{ulplx - u) < c;}, {ulply - u) < c;} are open disjoint neighbourhoods of x and y. For the converse we prove the apparently stronger statement that the separation property of (pJi E I is a consequence of E being a TI -space (i.e. the one point sets are closed). In fact, if x =f 0 and {x} is closed there exists a neighbourhood U of zero such that x ¢ U. By Proposition 1.7 there exist c; > 0 and finitely many indices iI, , in E I such that

{yIPil(y) < c;,

, Piney) < c;}

so for some i E {iI' ... , in} we have plx) ~ c;.

~

U,

D

1.9. Finest Locally Convex Topology. Let Ebe a vector space over IK. Among the topologies on E., which make E into a locally convex topological vector space., there is a finest one., namely the topology induced by the family of all seminorms on E. This topology is called the finest locally convex topology on E. An alternative way of describing this topology is by saying that the system of all absorbing absolutely convex sets is a base for the filter of neighbourhoods of zero, cf. 1.4. The finest locally convex topology is Hausdorff. In fact, let e E E\ {OJ be given. We choose an algebraic basis for E containing e and let qJ be the linear functional determined by qJ( e) = 1 and qJ being zero on the other vectors of the basis. Then p = IqJ I is a seminorm with pee) = 1, and the result follows from 1.8. Notice that every linear functional is contin uous in the finest locally convex topology. In Chapter 6 the finest locally convex topology will be used on the vector space of polynomials in one or more variables.

§2. Hahn-Banach Theorems

5

1.10. Exercise. Let E be a topological vector space, and let A, B, C, F s; E. (a) Show that A (b) Show that F

+ B is open in E if A is open and B is arbitrary. + C is closed in E if F is closed and C is compact.

1.11. Exercise. Let E be a topological vector space. Show that the interior of a convex set is convex. Show that if U is an absolutely convex neighbourhood of in E then its interior is absolutely convex. It follows that a locally convex topological vector space has a base for the filter of neighbourhoods of consisting of open absolutely convex sets.

°

°

1.12. Exercise. Show that a Hausdorff topological vector space is a regular topological space. (It is actually completely regular, but that is more difficult to prove.) 1.13. Exercise. Let E be a topological vector space and A s; E a nonempty and balanced subset. Then: (i) if A is open, A = {x E ElpA(X) < I}; (ii) if A is closed, A = {x E EI PA(X) ~ I}.

1.14. Exercise. Let P, q be two seminorms on a vector space E. Then if {x E Elp(x) ~ I} = {x E Elq(x) ~ I} it follows that P = q. 1.15. Exercise. Let the topology of the locally convex vector space E be induced by the family (pJiEI of seminorms, and let f be a linear functional on E. Then.f is continuous if and only if there exist C E JO, w[ and some finite subset J s; I such that I f(x) I ~ C • max{pi(x) liE J} for all x E E.

§2. Hahn-Banach Theorems One main result in the theory of locally convex topological vector spaces is the Hahn-Banach theorem about extensions of linear functionals. In the following we treat this and closely related results under the name of HahnBanach theorems. We recall that a hyperplane H in a vector space E over lK is a maximal proper linear subspace of E or, equivalently, a linear subspace of codimension one (i.e. dim E/H = 1). Another equivalent formulation is that a hyperplane is a set of the form qJ - I( {O}) for a linear functional qJ: E ~ lk\ not identically zero. Neither local convexity nor the Hausdorff separation property is needed in our first version of the Hahn-Banach theorem. However the existence of a nonempty open convex set A =F E is a strong implicit assumption on E.

6

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

2.1. Theorem (Geometric Version). Let E be a topological vector space over lK and let A be a nonempty open convex subset of E. If M is a linear subspace of E with A n M = 0, there exists a closed hyperplane H containing M with An H = 0. We first consider the case lK = IR. By Zorn's lemma there exists a maximal linear subspace H of E such that M ~ H and A n H = 0. Let C= H + > 0 AA. The sum of an open set and an arbitrary set is open, hence C is open, cf. Exercise 1.10. We now derive four properties of C and H by contradiction: PROOF.

U;.

(a) C n ( - C) = 0. In fact, if we assume x E C n (-C), we have x = h l + Alal = h 2 - A2a2 with hi E H, ai E A, Ai > 0, i = 1,2. By the convexity of A

which is impossible.

(b) H u C u ( - C) = E. In fact, if there exists x E E\(H u C u ( - C)) we define H = H + IRx, so H is a proper subspace of H. Furthermore A n H = 0 because yEA n H can be written y = h + AX with h E H and A =F (A n H = 0), and then x = (l/A)Y - (l/A)h E C u (-C), which is incompatible with the choice of x. Finally the existence of H is inconsistent with the maximality of H so (b) holds.

°

(c) Hn(Cu(-C))=0. In fact, if x E H n C then x = h + Aa with h E H, a E A and A > 0, but then a = (l/A)(x - h) E A n H, which is a contradiction. From (b) and (c) follows that H is the complement of the open set C u ( - C), hence closed.

(d) H is a hyperplane. If H is not a hyperplane there exists x E E\H such that H = H + IRx =F E. Without loss of generality we may assume x E C and we can choose y E ( - C)\H. The function f: [0, 1] ~ E defined by f(A) = (1 - A)X + Ay is continuous, sO.r - l(C) and.r - l( - C) are disjoint open subsets of [0, 1] and 1. Since [0" 1] is connected there exists containing" respectively" ct E ]0, 1[such thatf(ct) E H. But this implies y = (l/ct)(f(ct) - (1 - ct)x) E H, which is a contradiction. This finishes the proof of the real case. A complex vector space can be considered as a real vector space, and if H denotes a real closed hyperplane containing M and such that A n H = 0, then H n (iH) is a complex hyperplane with the desired properties. D

°

7

§2. Hahn-Banach Theorems

The following important criterion for continuity of a linear functional will be used several times. 2.2. Proposition. Let E be a topological vector space over IK, let qJ: E -+ IK be a nonzero linear functional and let H = qJ - 1( {O}) be the corresponding hyperplane. Then precisely one of the following two statements is true: (i) qJ is continuous and H is closed; (ii) qJ is discontinuous and H is dense.

PROOF. The closure H is a linear subspace of E. By the maximality of H we therefore have either H = H or H = E. If qJ is continuous then H = qJ - 1( {O}) is closed. Suppose next that H is closed. Let a E E\H be chosen such that qJ(a) = 1. By Proposition 1.2 there exists a balanced neighbourhood V of zero such that (a + V) n H = 0, and therefore qJ(V) is a balanced subset of lK such that 0 fll + qJ(V), hence qJ(V) f; {x E IK Ilx I < l}. It follows that IqJ(X) I < G for all x E GV, G > 0, so qJ is continuous at zero, and hence continuous. D 2.3. Theorem of Separation. Let E be a locally convex topological vector space over IK. Suppose F and C are disjoint nonempty convex subsets of E such that F is closed and C is compact. Then there exists a continuous linear functional qJ: E -+ IK such that sup Re qJ(x) < inf Re qJ(x). xeC

xeF

PROOF. Let us first suppose IK = ~, and consider the set B = F - C. Obviously B is convex, and using the compactness of C it may be seen that B is closed, cf. Exercise 1.10. Since F n C = 0 we have 0 €I B, so by 1.4 there exists an absolutely convex neighbourhood U of 0 such that U n B = 0. The interior V of U is an open absolutely convex neighbourhood (cf. Exercise 1.11) so A = B + V = B - V is a nonempty open convex set (1.10) such that 0 €I A. Since {O} is a linear subspace not intersecting A, there exists by Theorem 2.1 a closed hyperplane H with A n H = 0. Let qJ be a linear functional on E with H = qJ - 1( {O}). By 2.2, qJ is continuous. Now qJ(A) is a convex subset of ~, hence an interval, and since 0 €I qJ(A) we may assume qJ(A) f; ]0, 00 [. (If this is not the case we replace qJ by - qJ). We next claim inf qJ(x) > 0, xeB

which is equivalent to the assertion. If the contrary was true there exists a sequence (x n ) from B such that qJ(x n ) -+ O. Since V is absorbing there exists u E V with qJ(u) < 0, but X n + u E A so that qJ(x n) + qJ(u) > 0 for all n, which is in contradiction with qJ(x n) -+ O. In the case IK = C we consider E as a real vector space and find a ~-linear functional qJ: E -+ ~ as above. To finish the proof we notice that there exists precisely one C-linear functional t/J: E -+ C with Re t/J = qJ namely t/J(x) = qJ(x) - iqJ(ix), which is continuous since qJ is so. D

8

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

Applying the theorem to two one-point sets we find

2.4. Corollary. Let E be a locally convex Hausdorff topological vector space. For a, bEE, a =f= b, there exists a continuous linear functional f on E such that f(a) =f= f(b). We shall now treat the versions of the Hahn-Banach theorem which are called extension theorems. Although they may be derived from the geometric version, we give a direct proof using Zorn's lemma. The first extension theorem is purely algebraic and very useful in the theory of integral representations. It uses the following weakened form of the concept of a seminorm.

2.5. Definition. Let E be a vector space. A function p: E -+ IR is called sublinear if it has the following properties: (i) positive homogeneity: p(AX) = Ap(X) for A ~ 0, X E E; (ii) subadditivity: p(x + y) ~ p(x) + p(y) for x, y E E. A functionf: E

-+

IR is called dominated by p iff(x)

~

p(x) for all x E E.

2.6. Theorem (Extension Version). Let M be a linear subspace ofa real vector space E and let p: E -+ IR be a sublinear function. If f: M -+ IR is linear and dominated by p on M, there exists a linear extension 1: E -+ IR off, which is dominated by p. PROOF. We first show that it is always possible to perform one-dimensional extensions assuming M =f= E. Let e E E\M and define M' = span(M u {e}). Every element x' E M' has a unique representation as x' = x + te with x E M, r E IR. For every rx E IR the functional f~: M' -+ IR defined by f~(x + te) = f(x) + trx is a linear extension off. We shall see that rx may be chosen such thatf~ is dominated by p. By the subadditivity of p we get for all x, y E M

f(x)

+

f(y) = f(x

+ y)

~

p(x

+ y)

~

p(x - e)

or

f(x) - p(x - e)

~

p(e

+ y)

- f(y).

Defining

k = sup{f(x) - p(x - e)lx EM}, K = inf{p(e

+ y)

- f(y)ly EM},

we have - 00

< k

~

K
0 be such that x E AV. Then A-1 X E V n M and hence I f(x) I ~ A. This shows that the seminorm Pu determined by V (cf. 1.6) satisfies If(x)1 ~ Pu(x) for x EM. Let]be a linear extension offsatisfying 1](x)1 ~ Pu(x) for x E E. Then 1](x)1 ~ G for x E GV, which shows that]is continuous. D

2.10. If E denotes a topological vector space we denote by E' the vector space of continuous linear functionals on E, and E' is called the topological dual space, which is a linear subspace of the algebraic dual space E* of all linear functionals on E.

2.11. Exercise. Let E be a real vector space and p a sublinear function on E. Show that

p(x) = sup{f(x)lfEE*,f

~

p}.

2.12. Exercise. Let P1' ... , Pn: E -+ IR be sublinear functions on a real vector space E and let.f: E -+ IR be linear and satisfyingf(x) ~ P1(X) + ... + Pn(x) for x E E. Show that there exist linear functions f1' ... , fn: E -+ IR such that f = f1 + ... + fn and such that h is dominated by Pi for i = 1, ... , n. Hint: Consider the product space En. 2.13. Exercise. With the notation as in Theorem 2.6 we denote by A(f, E) the set of linear extensions]: E -+ IR offwhich are dominated by p. Clearly A (f, E) is convex. Show by a Zorn's lemma argument that A(f, E) has extreme points. Let Atf, E) such that

Xo E

E. Show that there exists an extreme point]o in

(For the notion of an extreme point see 2.5.1. The result of the exercise is due to Vincent-Smith (1966, private communication). For a generalization see Andenaes (1970).)

11

§3. Dual Pairs

§3. Dual Pairs Let r\J o = {O, 1,2, ... }, let E = ~~o be the vector space of real sequences s = (Sk)k~O and let F be the vector space of polynomials p(x) = Lk=O CkX" with real coefficients. Note that F can be identified with the subspace of sequences SEE with only finitely many nonzero terms. For sEE and p E F we can define 00

(s, p> =

L SkCk k=O

and (., . > is a bilinear mapping of E x F into ~, which clearly satisfies the axioms in the following definition, so E and F form a dual pair under ( ., .

>.

3.1. Definition. Let E and F be vector spaces over IK and (., . >: E x F -+ IK a bilinear form, i.e. separately linear. We say that E and F form a dual pair under (., . >if the following conditions hold: (i) For every e E E\{O} there existsf E F such that (e,f> =f= (ii) For every f E F\ {O} there exists e E E such that (e,f> =f=

o. o.

3.2. A locally convex Hausdorff topological vector space E and its topological dual space E' form a dual pair under the bilinear form (x, qJ> = qJ(x) for x E E, qJ E E'. The condition (ii) is clearly true and (i) follows from Corollary 2.4. A vector space E and its algebraic dual space E* form a dual pair under the bilinear form (x, qJ) = qJ(x). This example is a special case of the above example if E is equipped with the finest locally convex topology, cf. 1.9. We see below that every dual pair (E, F, (., . arises in the above way in the sense that there exist a topology 1J on E, such that E is a locally convex Hausdorff topological vector space, and an isomorphism j: F -+ E' such thatj(f)(e) = (e,f> for e E E,f E F. Such a topology 1J is called compatible with the duality between E and F. In general there exist many different topologies on E of this kind, and we will now define one, which turns out to be the coarsest compatible with the duality and therefore is called the weak topology.

»

3.3. Definition. Let E and F be a dual pair under (., . >. The weak topology a(E, F) on E is the topology induced by the family (Pf)feF of seminorms, where Pf(e) = I(e,f> I· Condition (i) of 3.1 implies that a(E, F) is Hausdorff, cf. 1.8. By reasons of symmetry there is also a weak topology a(F, E) on F.

3.4. Proposition. The topology a(E, F) is the coarsest of the topologies compatible with the duality between E and F. PROOF. If 1J is a topology compatible with the duality then e ~ (e,f> is 1J-continuous for all f E F, and so are the seminorms (Pf)feF. By 1.7 it

12

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

follows that aCE, F) is coarser than fJ. If E is equipped with the weak topology then e 1--+ <e,f> is a continuous linear functional on E for each f E F, and the mapping j: F -+ E' given by j(f)(e) = <e,f> is linear and one-to-one (condition (ii) of 3.1). To see thatj is onto we consider a aCE, F)-continuous linear functional qJ on E. By 1.7 there exists £ > 0 and .fl' ... ,In E F such that pflx) < £, i = 1, ... , n, implies IqJ(x) I ~ 1. This gives at once that

{x EEl <x,.h > = 0, i = 1, ... , n} ~ qJ Let us consider the linear mapping tjJ: E

tjJ(x) = «X,fl >,

-+

1(

{O} ).

(1)

/Kn defined by

... , <x,fn»'

XE

E.

The image tjJ(E) is a linear subspace of /Kn and the inclusion (1) implies that ip: tjJ(E) -+ /K is well defined by ijJ(tjJ(x)) = qJ(x), x E E. But a linear functional on a subspace of /Kn may be written n

L Ai Yb

ijJ(y) =

Y E tjJ(E)

~

/Kn,

i= 1

for a not necessarily unique vector (A l , ... , An) E /Kn, and this shows that qJ(x) = <x,f> withf = 1 Ai.h E F, hence j(f) = qJ. D

L?=

It is only slightly more difficult to show that there is also a finest topology on E compatible with the duality. This topology is called the Mackey topology and is denoted T(E, F), cf. Exercise 3.13. We now associate with each subset of one of the two vector spaces of a dual pair a subset of the other space of the pair, called the polar subset.

3.5. Definition. Let E and F be a dual pair under the polar subset A ° is given by

AO = {f

E

FIRe<e,f>

~

. For a subset A ~ E

1 for all e E A}.

For e E E the set {e}O = {f E FI Re<e,f> ~ I} is convex and closed in any topology, on F compatible with the duality. Therefore also

is ,-closed and convex. Furthermore 0 E AO.

3.6. The Bipolar Theorem. Let fJ be any topology on E compatible with the duality between E and F and let A ~ E. The bipolar set AOO = (AO)O is the smallest fJ-closed and convex subset of E" containing A u {O}. PROOF. From the above remark it follows that A is an fJ-closed and convex set containing A u {O}. To finish the proof we show that the existence of an fJ-closed convex set B containing A u {O} and a point e E AOO\B will lead to a contradiction. In fact, by the separation theorem (2.3) there exists an 00

13

§3. Dual Pairs

1]-continuous linear functional qJ: E

-+

IK and a number A E IR such that

Re qJ(e) < A < inf Re qJ(b). beB

Since 0 E B we have A < O. Ifj~'E F is such that qJ(x) = (x,f> for x E E we find

~~~ Re(b,~ f) < 1 < Re(e, ~ f)' The first inequality shows that (1/ A)f E A ° and the last inequality is then incompatible with e E A 00. D 3.7. Remark. If A is balanced we have

AO= {fEF/I(e,f>1 ~ IforalleEA}. This is often used as a definition of the (absolute) polar set. If A is a cone (i.e. AA £; A for all A ~ 0) we have

AO = {f E FI Re(e,f>

~

0 for all e E A},

which is a convex cone. With A £; E we also associate another convex cone Ai- £; F, which is closed in any topology on F compatible with the duality between E and F, namely Ai- = {fEFI(e,f> ~ Ofor alleE A}. Clearly Ai-£; - A ° and if E and F are real vector spaces and if A is a cone then Ai- = -AO. For a set A containing 0 the bipolar theorem states that A 00 is the 1]-closed convex hull of A. Using translations we therefore have the following consequence of the bipolar theorem: 3.8. Proposition. The closed convex hull of a subset of E is the same for all topologies on E compatible with a given duality. If E is a finite dimensional vector space over IK, hence isomorphic with IK n where n is the dimension of E, there is exactly one topology on E compatible with the duality between E and E*. More generally there is exactly one Hausdorff topology on E such that E is a topological vector space. We will refer to this topology as the canonical topology of E. These assertions are contained in the following result. 3.9. Proposition. Let E be a finite dimensional subspace of a Hausdorff topological vector space F. Then E is closed in F, and any algebraic isomorphism qJ: IK n -+ E (n = dim(E)) is a homeomorphism, when IK n is equipped with the topology generated by the euclidean norm.

14

1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs

We first show by induction that any isomorphism qJ: IK n -+ E is a homeomorphism. For n = 1 we put qJ(l) = e. The continuity of scalar multiplication implies that qJ: A 1--+ Ae is continuous from IK to E. The inverse qJ - 1 is a linear functional on E, and its kernel is the hyperplane {O}, which is closed since E is Hausdorff. By 2.2 it follows that qJ - 1 is continuous. Let us assume that the above statement is true for all dimensions less than n and let qJ: IK n -+ E be an algebraic isomorphism. As before the continuity of the algebraic operations shows that qJ is continuous. To see that n qJ - 1 : E -+ IK is continuous it suffices to prove that each linear functional on E is continuous. To get a contradiction let us assume that t/J: E -+ IK is a discontinuous linear functional and put H = t/J - 1({O}). Then H is a (n - 1)dimensional hyperplane, which is dense in E by 2.2. Let 11·11 be the euclidean norm (or any norm) on H. By the induction hypothesis the norm topology on H coincides with the topology induced from E, so there exists an open set V in E such that V n H = {xEHlllxll < I}. PROOF.

Since H is dense in E and V is open, we have V n H = V, where the closures are in E. But the set {x EHlllxli ~ I} is compact in H, hence in E and in particular closed in E, so we get V ~

V = V n H ~ {x EHlllxli ~ I}. Since V is absorbing in E we get E = H. By this absurdity

qJ is indeed a homeomorphism. We finally show that E is closed in F. If this is not true there exists x E E\E. Then E = span(E u {x}) is a (n + I)-dimensional space. If et, ... , en is an algebraic basis for E, then qJ: IK n + 1 -+ E given by qJ(A t , .•. , An' A) = 1 Ai ei + AX is an algebraic isomorphism, hence a homeomorphism. It follows that E is closed in E, hence x E EnE = E, which is a contradiction.

Li=

D

.>.

3.10. Exercise. Let E and F be a dual pair under (., Then the weak topology (J(E, F) is the coarsest topology on E for which the mappings e 1--+ (e,f> are continuous when .f ranges over F. 3.11. Exercise (Theorem of Alaoglu-Bourbaki). Let E be a locally convex Hausdorff topological vector space with topological dual space E' and let V be a neighbourhood of zero in E. Show that VO is (J(E', E)-compact. Hint: Show that for x E E there exists A > 0 such that I(x, .f> I ~ A for all

f

E

vo.

3.12. Exercise. Let E, F be a dual pair under (., . >and let t1 be a topology on E compatible with the duality. Let V be a closed, absolutely convex neighbourhood of zero in E and let Pu be the seminorm determined by V, cf. 1.6. Show that XE E. Pu(x) = sup{ I(x,f> f E Va},

II

15

Notes and Remarks

3.13. Exercise (Theorem of Mackey-Arens). Let E, F be a dual pair under (., .), and let d be the family of all absolutely convex and u(F, E)-compact subsets of F. For A E t91 we define

IlellA

= sup{ I(e,f)

II f

E

A},

e E E.

Show that II·IIA is a seminorm on E. Use 3.11 and 3.12 to show that if fJ is a topology on E compatible with the duality then fJ is induced by some subfamily of (11·11 A)A E.>4· Show finally that the topology induced by the family (11·IIA)AEd is the Mackey topology, i.e. the finest topology on E compatible with the duality.

Notes and Remarks In the period up to the 1940's most results in functional analysis were about normed spaces. The development of the theory of distributions of Schwartz was one main motivation for a study of general spaces, since the basic spaces of test functions and distributions are nonnormable in their natural topology. Today locally convex Hausdorff topological vector spaces are a natural frame for many theories and problems in functional analysis, e.g. the theory of integral representations, which we shall discuss in the next chapter. For historical information on the theory of topological vector spaces we refer the reader to the book by Dieudonne (1981).

CHAPTER 2

Radon Measures and Integral Representations

§1. Introduction to Radon Measures on Hausdorff Spaces It is well known that the pure set-theoretical theory of measure and integration has its limitations, and many interesting results need a topological frame because measure spaces without an underlying "nice" topological structure may be very pathological. In classical analysis this difficulty was overcome by introducing the theory of Radon measures on locally compact spaces. On these spaces there is a particularly important one-to-one relationship between Radon measures and certain linear functionals (see below) which in many treatments on analysis leads to the definition, that a Radon measure is a linear functional with certain properties. Another branch of mathematics with a need for a highly developed measure theory is probability theory. Here the class of locally compact spaces turned out to be far too narrow, partly due to the fact that an infinite dimensional topological vector space never can be locally compact. For example, it was found that the class of polish spaces (i.e. separable and completely metrizable spaces) was much more appropriate for probabilistic purposes. Later on it became clear that a very satisfactory theory of Radon measures can be developed on arbitrary Hausdorff spaces. This has been done, for example, in L. Schwartz's monograph (1973). We shall follow an approach to Radon measure theory which has been initiated by Kisynski and developed by Tops~e. It deviates, for example, from the Schwartz-Bourbaki theory in working only with inner approximation, but we hope to show that it gives an easy and elegant access to the main results.

§1. Introduction to Radon Measures on Hausdorff Spaces

17

In the following let X denote an arbitrary Hausdorff space. The natural a-algebra on which the measures considered will be defined will always be the a-algebra f!4 = f!4(X) of all Borel subsets of X, i.e. the a-algebra generated by the open subsets of X. In our terminology a measure will always be nonnegative; a measure defined on f!4(X) will be called a Borel measure on X. Later on we also need to consider a-additive functions on f!4(X) which may assume negative values, these functions will be called signed measures.

1.1. Definition. A Radon measure J1 on the Hausdorff space X is a Borel measure on X satisfying (i) Jl(C) < 00 for each compact subset C £ X, (ii) Jl(B) = sup{Jl(C) I C £ B, C compact} for each B E f!4(X). The set of all Radon measures on X is denoted M +(X).

Remark. Many authors require a Radon measure to be locally finite, i.e. each point has an open neighbourhood with finite measure. There are good reasons for not having this condition as part of the definition, see Notes and Remarks at the end of this chapter. A finite Radon measure Jl (i.e. Jl(X) < (0) satisfies for B E f!4(X) Jl(B) = inf{Jl(G)IB £ G, G open} as is easily seen by considering the property (ii) for Be. However for arbitrary Radon measures this need not be true as is shown by Exercise 1.30 below. Let .ff = .ff(X) denote the family of all compact subsets of X. Clearly the restriction to .ff of a Radon measure Jl is a set function

A: .ff

-+

[0, oo[

satisfying the axioms of a Radon content below.

1.2. Definition. A Radon content is a set function A:.ff -+ [0, oo[ which satisfies (1) A(C 2 ) - A(C 1 ) = sup{A(C)IC £ C 2 \C 1 , CE.ff} for all C 1 , C 2 E.ff with C 1 £ C 2 • The key result in our approach to Radon measure theory is the extension theorem (1.4) below, the proof of which will need the following lemma.

1.3. Lemma. A Radon content A has the following properties: (i) A(C t ) ~ A(C 2 ),for all Cf, C 2 E ~ C t £ C 2 , i.e. A is monotone. (ii) A(C t U C 2 ) + A(C t n C 2 ) = A(C 1 ) + A(C 2 )'1 i.e. A is modular. (iii) lj'a net (Ca)(J. E A in.ff is decreasing with C = na C a then A( C) = lima A(Ca) = infa A( C a). In particular for a decreasing sequence C 1 ;2 C 2 ;2 . . . of compact sets we have lim n _ oo A(C n) = A(n:=l Cn).

18

2. Radon Measures and Integral Representations

(i) as well as A(0) = 0 is obvious. (ii) We have (C 1 u C 2 )\C 2 = C 1 \(C 1 n C 2 ) and therefore

PROOF.

A(C 1 u C 2 )

A(C 2 ) = A(C 1)

-

A(C 1 n C 2 )

-

as an immediate consequence of (1). (iii) Assume that b := in(iA(C a) - A(C» > O. We choose a fixed set C ao and C' £ Cao \ C, C' E .ff such that

A(Cao) - A( C) - A(C') < b. Now na~ao(C' n C a) = 0 and therefore C' n Cal = 0 for some Cal £ Cao since C' is compact and (Ca)aE A is decreasing. From (ii) we get

A(C' u Ca) = A(C') < A(C')

+ A(Ca) ~ A(Cao ) + A(C) + b

o

implying A(C a) - A(C) < J, a contradiction.

1.4. Theorem. Any Radon content on a Hausdorffspace has a unique extension to a Radon measure. PROOF. Let A be a Radon content on X. We define for any subset A £ X the inner measure by A*(A):= sup{A(C)IC £ A, C E.ff}

and have to show that J1 := A* I~ is a measure. Of course A* is an extension of A, but it may assume the value + 00, if Ais unbounded. In a certain analogy with Caratheodory's famous abstract measure extension theorem we consider the set system

.91:= {A

£

X/A*(C n A)

+

A*(C n A C ) = A*(C) for all C

E

.ff},

and we will show that d is a a-algebra containing ~, on which the restriction of A* is a-additive. From the very definition d is closed under complements and contains the empty set. The defining property (1) of a Radon content shows that .91 even contains all open subsets of X. Let AI' A 2 E d be disjoint and let C 1 £ A b C 2 £ A 2 be compact. Then the modularity of A gives

A(C 1 )

+

A(C 2 ) = A(C 1 u C 2 ) ~ A*(A 1 u A 2 )

and hence

A*(A 1 )

+

A*(A 2 ) ~ A*(A 1 u A 2 ),

i.e. A* is "superadditive". As a consequence d may also be written as

d = {A £ X/A*(C n A)

+

A*(C n A C ) ~ A*(C) for all C

E

.ff}.

Now let a sequence AI' A 2 , ••• E.91 be given and fix C E.ff as well as o. Then there exist compact sets K j £ C n A j and L j £ C n Aj such that

G

>

j

= 1,2, ....

(2)

19

§1. Introduction to Radon Measures on Hausdorff Spaces

From the modularity of A we get

A(V: K

i)

+ A(K n + 1n

/)1 Ki )

n

~Cn

=

A(VI K

i)

+ A(K n +

I)

(3)

as well as

We have

and

I n := L n + hence

K

n

and

1 U

j-= 1

Li

(n

A~+ I)'

Aj u

j= 1

In are disjoint compact subsets of C, so that A(K n ) + A(I n ) ~ A(C).

(5)

Adding the equalities (3) and (4) and inserting (2) and (5) give

ACV: K

i)

+ AC6: Li ) =

A(~I K i ) - A(K n )

+

-

A(OI L

i)

I)

+ A(K n + + A(Ln +

A(I n )

(6)

~ A() Ki ) + A(.n Li ) - 2n~ )=1 )=1 Ifwe add (6) over n

I)

l'

= 1,2, ... , N - 1 and use (2) for j = 1 we get

A(~I K i )

+ A(6 Li )

~ A(C) -

/;-

JI ~.

(7)

Put A:= Uf=1 A j ; then A(U7=1 K j ) ~ A*(C n A) for all Nand

A(C\ Li ) =

!~n:, A(b Li ) ~ A*(C n

A

C )

(8)

by Lemma 1.3(iii), hence letting N tend to infinity in (7) gives

A*(C n A)

+ A*(C n

A

C )

~

A(C) -

B,

and since this holds for all B > 0, we have in fact shown A E or", hence d is a a-algebra containing the open sets and therefore the Borel sets. Let us now furthermore assume that the sets A l' A 2' ... E d are pairwise disjoint and that C £ A. Then lim N _ oo A(n7=1 L j ) = 0 by (8), and taking again the limit in (7) gives

Jc(C) -

B

~ ~~~ A(~I Ki)= ~~ itl A(K) = JI A(K) ~ i~1 A*(A)

20

for all

2. Radon Measures and Integral Representations

G

> O. Letting G -+ 0 we find 00

A*(A) = sup{A(C)IC £ A, C

E

L A*(A

$'} ~

j ),

j= 1

and since the reverse inequality is obvious by the superadditivity of A* we have that A* I.rtf is a measure, thus finishing our proof. D The result we are now going to prove is a kind of monotone convergence theorem for Radon measures. The usual form of this theorem on general measure spaces deals with an increasing sequence of nonnegative measurable functions; however, if the underlying measure is a Radon measure and if the functions to be integrated are lower semicontinuous (i.e. {.f > t} is open for all t E [R), then the sequence may be replaced by an arbitrary increasing net of functions, as we shall see. In the sequel we shall make repeated use of the obvious inequality 1

00

o ~ J.:= 2" i~l l u >i/2"j ~ f

(9)

being valid for arbitrary functions f with values in [0, 00]. Iff is finite the infinite series in (9) reduces to a finite sum (pointwise) and f - .in ~ 1/2n • Note thatfn increases tofalso iff assumes the value 00. Let us mention that the family of all lower semicontinuous functions is closed under finite sums, multiplication with a nonnegative constant, and that the supremum of an arbitrary subfamily of these functions is still lower semicontinuous. Noting finally that an indicator functionf = I G is lower semicontinuous if and only if G is open, we see that the functionsfn defined in (9) are lower semicontinuous if fis.

1.5. Theorem. Let J1 be a Radon measure on the Hausdorffspace X. Then the .following holds: (i) If a net (Ga)aEA of open subsets of X is increasing with

Ua Ga=

G then

J1(G) = sup J1(G a) = lim J1(G a). (ii) Ifa net (fa)(J.EA oflower semicontinuousfunctions X with sUPa j~ = f then

f

f dJl =

f

s~p f~ dJl = li~

ffa

-+

[0, 00] is increasing

dJlo

PROOF. (i) Let C £ G be compact. Then finitely many Gal' ... , Gak cover C and by assumption there is some CXo such that Gal U ••. U Gak £ Gao' implying J1(C) ~ J1(G ao ) ~ SUPa J1(G a) and therefore

J1(G) = sup{J1(C)I C £ G, C The reverse inequality is trivial.

E

$'}

~

sup J1(G a).

21

§l. Introduction to Radon Measures on Hausdorff Spaces

(ii) For every t E IR the open sets {fa> t} increase to {f > t}. Using the functions.in and the correspondingfa,n as defined in (9) we find

If~d~

=

;n ~~({f > ;n}) = ;n ~li~~({fa > ;n})

=

li~ ;n ~ ~ ({fa > ;n}) = li~ If~. nd~,

where the interchange of limits is justified, both limits being suprema, and using this device once more we get

I

f d~ = s~p I fn d~ = s~p s~p I f~.

n

d~

= sup sup Ifa,n dfl = sup I fa dfl, a

n

a

D

applying, of course, the usual monotone convergence theorem.

1.6. Remark. Theorem 1.5 can be applied to an upwards filtering family A of sets or functions by defining an increasing net in the following way: The index set and the mapping of the net will be A and the identical mapping. A Borel measure satisfying property (i) of the above theorem is usually called a r-smooth measure. The class of these measures is in general larger than the class of Radon measures, however, for finite Borel measures on locally compact spaces the two notions coincide. The generalized monotone convergence theorem expressed as property (ii) of the above theorem uses only the r-smoothness of the underlying Radon measure and therefore remains valid for r-smooth measures as well, see Top;¢>e (1970) and Varadarajan (1965). We shall need in the following the notion of restriction ofa Radon measure to a Borel subset. If X is a Hausdorff space and B E 81(X), then B is again a Hausdorff space with respect to the trace topology {B n GIG open in X} and it is easy to see that the Borel subsets of B are given by

81(B) = {B n AlA

E

81(X)} = {D

so that in fact 81(B) £ 81(X). For fl

E

E

81(X)ID £ B}

M +(X) we now define

fll B: 81(B)

~

[0, 00]

as the restriction of fl to 81(B)'l i.e. (fll B)(A) := fl(A) for A mediately seen that fll B is again a Radon measure.

E

81(B). It is im-

1.7. Proposition. Let J1 be a Radon measure on X. lfthefunctionf: X is Borel measurable, then

If d~

= sup Ke,*,"

r f d~,

JK

~

[0,00]

(10)

22 and

2. Radon Measures and Integral Representations

if f:

X

-+ [o~

oo[ is continuous then v: &I(X) v(B):=

-+ [o~

00] defined by

{! d,u

is again a Radon measure. The measure v is often denoted fJ1 or f dJ1. PROOF. If I = I B for some B E &I(X), then (10) follows from the definition of a Radon measure. It is obvious that (10) remains true iff'is an elementary measurable nonnegative function, i.e. f = = I CXi 1Bi with pairwise disjoint Borel sets Bf, ... , Bn and CXI' ••• , CXn ~ o. But it is well known that an arbitrary Borel measurable I ~ 0 is the pointwise limit of some increasing sequence of elementary functions, so that the usual monotone convergence theorem and the possibility of interchanging two suprema give (10) in the general case also. Let now f: X -+ IR+ be continuous and v(B) = SBI dj1, BE &I(X). Obviously, v is finite on compact sets. Applying (10) to the restrictions J1IB and.flB we find

Ii

{!

{{!

d,u = su p

d,uIK

E

Jf; K

~ B}

0

1.8. Let j1 be a Radon measure on X and consider the family ~ of all open J1-zero sets in X. The system of all finite unions of sets in ~ filters upwards to the union G of all sets in ~ and J1(G) = 0 by Theorem 1.5. The open set G is therefore maximal in ~ and its complement is called the support of j1 or abbreviated supp(J1). It is immediate that supp(j1) is closed and that sUPP(J1) = {x

E

X IJ1(U) > 0 for each open set U such that x

E

U}.

Particularly simple examples of Radon measures are those with a finite support which we will call molecular measures, and among these are the one-point or Dirac measures Gx defined by Gx ( {x}) = 1 and Gx ( {x }C) = o. Of course supp(Gx ) = {x} and if J1 = I CXiG Xi is a molecular measure with Xi Xj for i j, then sUPP(j1) = {XdCXi > O}. The set of molecular measures is denoted Mol+(X).

+

+

I?=

In the usual set-theoretical measure theory, as well as in the theory of Radon measures, the notion of a product measure is of central importance. In the latter case we are immediately confronted with the following problem: Let X and Y be two Hausdorff spaces; then the product of the two u-algebras of Borel sets~ usually denoted &I(X) ® &I(Y)~ is by definition the smallest u-algebra on X x Y rendering the two canonical projections 1Cx: X x Y -+ X and1Cy: X x Y -+ Ymeasurable,i.e.&I(X) ® &I(Y)istheu-algebragenerated by 1Cx I (&I(X» U 1Cf 1(&I( Y». By definition of the product topology these two projections are continuous on X x Y and therefore Borel measurable, so that always

&I(X) ® &I(Y) £ &I(X x Y).

23

§l. Introduction to Radon Measures on Hausdorff Spaces

On "nice" spaces we even have equality of these two a-algebras on X x Y, but this need not always hold, see the exercises below. Our next goal will be to show existence and uniqueness of the product of two arbitrary Radon measures. This stands in some contrast to set-theoretical measure theory where usually a-finiteness of the measures is required in order to guarantee a uniquely determined product measure. We begin with a lemma.

1.9. Lemma. Let Z be a Hausdorffspace and let stl be an algebra ofsubsets of Z containing a base for the topology. If A: .91 --+ [0, 00 [ is finitely additive then A: %(Z) --+ [0, oo[ defined by ,1(C):= inf{A(G)I C £ G, G open, G E d} is a Radon content on Z. PROOF. Let C £ Z be compact, then every point x E C has an open neighbourhood Gx E d. Finitely many of these neighbourhoods cover C and their union is still in d. Hence ,1( C) is certainly finite. Now let two compact sets C 1 £ C 2 be given. For B > 0 there is an open set G 1 ;2 C b G 1 Ed such that A(G 1 ) - ,1(C 1 ) < B. The set C:= C 2 n G1 is compact, too, allowing us to choose a further open set G E .91, G ;2 C with A(G) - ,1(C) < B. Of course, C 2 £ G U G 1 E .91 so that ,1(C 2 ) ~ A(G) + A(G 1) and therefore ,1(C 2 ) - ,1(C 1) ~ A(G) + A(G 1 ) + B - A(G 1 ) < ,1(C) + 2B. Hence

,1(C 2 )

-

,1(C 1 )

~

sup{,1(C) I C £ C 2 \C 1 , C

E

%}.

The reverse inequality will follow immediately if we can show that A is additive on disjoint compact sets. Therefore let K, L E % with K n L = 0 be given. One direction, namely

,1(K u L)

~

,1(K)

+ ,1(L)

is obvious, so it remains to be shown that for arbitrary

,1(K)

+ ,1(L)

~

,1(K u L)

B

> 0

+ B.

By definition there is an open set WEd containing K u L such that A(W) - ,1(K u L) < B. The assumption made on the algebra .91 implies that K and L may be separated by open sets G, H belonging to .91, i.e. we have K £G,

L £H,

Gn H = 0.

Hence

,1(K)

+ ,1(L)

~

A(G n W)

+ A(H n

W)

= A«G u H) n W) ~

thus finishing the proof.

A(W) < ,1(K u L)

+ B, D

24

2. Radon Measures and Integral Representations

Later on we shall need existence and unicity of certain Radon measures on the product of two Hausdorff spaces X and Y not only for the product of two measures, but also for so-called Radon bimeasures. If(X, d) and (Y,!!l) are just two measurable spaces (without an underlying topological structure) then a bimeasure is by definition a function : .91 x !!l

-+

[0,

00]

such that for fixed A E .91 the partial function B ~ (A, B) is a measure on !!l and for fixed B E !!l the function A ~(A, B) is a measure on d. Obviously, if K is a measure on .91 @ !!l, then (A, B) ~ K(A x B) is a bimeasure, but in general not even a bounded bimeasure is induced in this way, cf. Exercise 1.31. Our next result will, however, show that for Radon bimeasures such pathologies do not exist, where by definition is a Radon bimeasure if is a bimeasure defined on !!l(X) x !!l( Y) such that (K, L) < 00 for all compact sets K, Land (A, B) = sup{(K, L)IA ;2 K E g(X), B;2 L E g(y)} for all Borel sets A and B. 1.10. Theorem. Let X and Y be two Hausdorff spaces and let : !!l(X) x !!l(Y) -+ [0, 00] denote a Radon bimeasure. Then there is a uniquely determined Radon measure K on X x Y with the property

(K, L) = K(K x L)

.for all

K

E

g(X),

L

E

g(y).

Furthermore, the equality (A, B) = K(A x B) holds for all Borel sets A

E

!!l(X), B

E

!!l( Y).

PROOF. Denote Z:= X X Y and let .91 be the algebra generated by the "measurable rectangles" A x B, where A E !!l(X) and BE !!l(Y). This algebra contains, of course, the products of open sets in X (resp. Y) and therefore a base for the topology on Z. It is easy to see that there is a uniquely determined finitely additive set function A on .91 fulfilling

A(A x B) = (A, B)

for all

A

E

!!l(X)

and

B E !!l( Y).

Let us now first assume that (X, Y) < 00. Then we may apply Lemma 1.9 which, combined with the extension theorem 1.4, shows the existence of a Radon measure K on Z such that K(C)

= inf{A(G)IC

£

G E .91, G open}

for each compact set C £ Z. If C = K x L is the product of two compact sets K £ X, L £ Y, then C E .91 and K(K x L)

~

A(K x L) = (K, L)

by monotonicity of A. On the other hand, we may use the two finite Radon measures ,u(A):= (A, Y) on X and v(B):= (X, B) on Y to provide us

25

§ 1. Introduction to Radon Measures on Hausdorff Spaces

with open sets G

;2

K, H

;2

L such that J.1(G\K)


if f(x)

= g(x) = 0,

if f(x)

+ g(x) >

0,

0,

if.f(x) = g(x) = 0,

f, h"

~

g and h'

T(h)

~

T+(f)

+ h" = h, implying + T+(g)

so that finally T+ is additive on C +(X). We put T- := T+ - T which also is additive, nonnegative and positively homogeneous on C + (X). By Corollary 2.3 there are two Radon measures J1b J12 on X such that

41

§2. The Riesz Representation Theorem

implying for f E C(X) TC{) = T(f+) - T(f-) = T+(f+) - T-(f+) - [T+(f-) - T-(f-)]

= I f+ d(J.11 - J.12) - I f- d(J.11 - J.12) J1

= II dJ.1,

:=

J1l - J12

so that T = ~. By Corollary 2.3 we also get immediately that there is only one signed Radon measure J1 with this property. Let J1 = J1+ - J1- denote the Jordan-Hahn decomposition of J1, i.e. J1 +(B) = J1(B n D) and J1- (B) = J1(B n DC) where D E &leX) is chosen in such a way that both J1 + and J1- are nonnegative (see, for example, Billingsley (1979, p. 373)). Then J1+ and J1- are Radon measures and furthermore

I I dJ.11

=

sup{Ih

~

I f dJ.1+

dJ.1+ - I h dJ.1-IO

~ h ~ f, h E C(X)}

for all.{EC+(X), hence J1l ~ J1+ by Exercise 1.27 and similarly J12 ~ J1-. Consequently we get J1l (DC) = 0 = J12(D) and therefore J1l = J1 +, J12 = J1as well as 1IJ111 = J1l (D) + J12(D C). We still have to show the reverse inequality II TJlII ~ 1IJ111. Given e > 0 we choose compact sets K+ £; D, K- £; DC such that

By Urysohn's lemma there is a continuous functionf: X thatflK+ == 1 andflK- == -1, implying

If

f dJ.11

~ J.11(K+) + J.1iK-) -

2/:

~ 11J.111 -

~

[ -1,1] such

4/:.

This shows II TJlII ~ 1IJ111 and finishes the proof of the theorem.

D

We shall now consider the case where X is locally compact. For a given compact subset K £; X we denote by C K(X) the vector space of all continuous functions on X whose support is contained in K; of course, CK(X) is a Banach space with respect to the supremum norm and CC(X)

=

U

CK(X).

K E Jf"(X)

A linear functional T on CC(X) is called continuous if all the restrictions TIC K(X) are continuous in the usual sense.

2.5. Theorem. For a locally compact Hausdorff space X the continuous linear jiinctionals T on CC(X) are in a bijective linear relation with the signed

42

2. Radon Measures and Integral Representations

Radon measures J1 on X via the natural formula T(.f) =

f f dJ1.

If J1 is a signed Radon measure, i.e. J1 = J1l - J12 for Radon measures J1l' J12' then TCf) = f dJ1 := f dJ1l .f dJ12 is well defined and continuous in the above sense; the unicity of the representing measure J1 is obvious by Corollary 2.3. Now let a continuous linear functional T: CC(X) ---+ IR be given. Imitating the proof of Theorem 2.4 we define for f E CC+ (X) PROOF.

J

J

J

T+(f):= sup{T(h)l.f

~

hE CC+(X)}.

Any function h occurring here belongs to Csupp(flX), implying that for some constant a > 0 IT(h) I ~ allhll ~ all.!'II, hence

o~

T+(j)
0 there exists a compact subset K elg(x)l.for x E X\K.

£;

X such that If(x)1

~

2.6. Definition. A convex cone C £; C +(X) of nonnegative continuous functions is called an adapted cone if: (i) For every x (ii) For everY.f

E E

X there exists.f E C such that.f(x) > O. C there exists gEe such that.f E o(g).

A linear subspace V

£;

C(X) is called an adapted space if:

(iii) V = V+ - V+, where V+ = V n C +(X). (iv) V+ is an adapted cone.

If.f E CC+(X) then.f E o(f) and for.f E C1(X) we have.f E o(J]), which shows that CC(X) and CO(X) are adapted spaces. If p and q are two real

43

§2. The Riesz Representation Theorem

polynomials in one variable then p E o(q) if and only if deg(p) < deg(q). It follows easily that the space of polynomials in one variable is an adapted space of continuous functions on X = IR. Similarly the space of polynomials in k variables is adapted on X = IRk. The main property of adapted spaces is given in the following: 2.7. Theorem. Let V be an adapted space of continuous functions on a locally compact space X. For every positive linear functional L: V ~ IR there exists a Radon measure J1 on X such that V £; ,pl(J1) and

PROOF.

L(f)

= ff

{f

C(X)llfl ~ g for some g E V+}.

for all .f E V.

dp

We define

v=

E

Then V is a subspace of C(X) containing V, and a simple compactness argument combined with (i) shows that CC(X) £; V. We claim that V= V+ + Jt: In fact, iff E Ii and g E V+ is such that If I ~ g, thenf = (g + f) + (-g) shows the assertion. By Corollary 1.2.7 it follows that L can be extended to a positive linear functional L: V ~ IR, and by the Riesz representation theorem there exists a Radon measure J1 on X representing [ ICC(X). For g E V+ and qJ E CC(X) satisfying 0 ~ qJ ~ g we have

f qJ dp.

L(g) = L(g) ;;;; L(qJ) = By U rysohn's lemma the family ff to g, so by Theorem 1.5

f

g dp

=

:=

{qJ

CC(X) I0 ~ qJ ~ g} filters upwards

E

sup{f qJ dpl qJ E.9F} ~ L(g) < 00, J

hence g E !fJl(J1). To see that g dJ1 = L(g) we choose h E V+ such that g E o(h). Let e > 0 be given. There exists a compact set K £; X such that ~

g(x) We choose qJ

E

eh(x)

CC(X) such that Ix

x

for ~

o~ g -

qJ

~

gqJ

E

X\K.

1 and find ~

eh,

hence

o ~ L(g) -

l(gqJ) ~ eL(h),

or

L(g)

~

f

gqJ dp

+ eL(h) ~

f

g dp

+ eL(h),

44

2. Radon Measures and Integral Representations

which suffices since e > 0 is arbitrary and independent of h. The equality

L(g) =

f

g d/l

D

extends clearly from V+ to V.

2.8. Exercise. Let X be locally compact. Show that the dual (Co(X))' of the Banach space CO(X) equals Mb(X), the space of all finite signed Radon measures on X (which is again a Banach space with respect to the total variation norm) in the sense, that there exists a linear bijective isometry from Mb(X) onto (CO(X))'. Hint: Use the one-point compactification of X. 2.9. Exercise. Show that the following conditions are equivalent for a locally compact space X: (i) X is a-compact, i.e. X is a countable union of compact sets. (ii) There is a strictly positive function.r E CO(X). (iii) There is a function f E C(X) with f(x) ---+ 00 for x ---+ 00, i.e. such that tr ~ a} is compact for all a E IR. (iv) C(X) is an adapted space.

2.10. Exercise. Try to find a finite signed Radon measure J1 on IR such that supp(J1+) = sUPP(J1-) composition of J1.

= IR~ where J1 = J1+ - J1- is the Jordan-Hahn de-

2.11. Exercise. Let X be locally compact and consider on CC(X) the following four families (Pi)ielj,j

XI0 =f

II

=

12

= f(X),

{A

£;

= 1, ... ,4, of seminorms: PAC.{):= max 1.{(x)l,

A finite},

xeA

PK(f):= max If(x)l, xeK

13 = {oo},

Poo(f)

:=

sup I f(x) I, xeX

{pili

E

I 4 } = {pip is a seminorm on CC(X) whose restriction to CK(X) is continuous for all K

E

f(X)},

where CK(X) := {f E CC(X) Isupp(f) £; K} is considered as a Banach space with respect to the supremum norm. Let (!) l' ... , (!) 4 denote the corresponding locally convex topologies on CC(X) (which are all Hausdorff). Show that the topological duals of CC(X) with respect to these four topologies are given (by natural identification) in the following way: (CC(X), (D.)' = {J1

E

M(X)lsupp(J1) is finite},

(CC(X), (D2)' = {J1

E

M(X)lsupp(J1) is compact},

(CC(X), (D3)' = {J1

E

M(X) 111J111 < oo},

(CC(X), (D4)'

=

M(X),

~3.

45

Weak Convergence of Finite Radon Measures

where supp(J1):= supp(,u+) decomposition of ,u.

U

supp(J1-) if J1 = J1+ - J1- is the Jordan-Hahn

2.12. Exercise. Let V be an adapted space of continuous functions on a locally compact space X, let F

X be a closed subset and put

£;

V~ = {f E V I f(x) ~ 0 for all x E F}.

Show that any linear functional L: V be represented as

L(f) =

ff

---+

IR which is nonnegative on V~ can for

dp

f

E

V,

where ,u E M +(X) is supported by F.

2.13. Exercise. Let X, Y be locally compact spaces and f: X ---+ Y a continuous surjective mapping such that f-l(K) is compact in X for each compact subset K £; Y. Show that each v E M +(Y) is of the form J11 for some ,u E M +(X). Hint: Use Corollary 1.2.7 and the Riesz representation theorem. 2.14. Exercise. Let X and Y be locally compact spaces and T: CC(X) x CC(Y) ---+ IR a bilinear mapping which is positive in the sense that T(f, g) ~ 0 if.f E CC+ (X) and g E CC+ (Y). Show that

T(f, g) = for some uniquely determined J1

E

ff

® 9 dp

M + (X x Y).

§3. Weak Convergence of Finite Radon Measures The theory of weak convergence of finite Radon measures is a well-developed theory which is of great importance in probability theory, in particular when dealing with stochastic processes. We will later need only a very few basic facts which we are going to develop in this section. Let X be a Hausdorff space and denote by M~(X) the set of all finite Radon measures on X, i.e. all Radon measures J1 with J1(X) < 00. The weak topology on M~ (X) is the coarsest topology such that the functions J1 ~ f dJ1 become lower semicontinuous for every bounded lower semicontinuous .f: X ---+ IR. The family of sets

J

Gf,I:=

{p

E

M:(X)! ff

dp >

t}

is a subbase for the weak topology when f ranges over the bounded lower semicontinuous functions on X and t E IR. The following result is part of the so-called portmanteau theorem (cf. Tops~e 1970, Theorem 8.1).

46

2. Radon Measures and Integral Representations

3.1. Theorem. For J1 E M~(X) and a net (J1(l)(lEA in M~(X) the following properties are equivalent: (i) (ii) (iii) (iv) (v)

J1(l

~

lim lim lim lim

J1 weakly, i.e. in the weak topology; inf J1(l(G) ~ J1(G)for all open G £; X and lim J1(l(X) = J1(X); sup J1(l(F) ~ J1(F)for all closed F £; X and lim J1(l(X) = J1(X); inf Jf dJ1(l ~ Jf dJ1for all bounded lower semicontinuousf: X ~ ~; sup Jf dJ1(l ~ Jf dJ1for all bounded upper semicontinuousf: X -+ IR.

If (i)-(v) are fulfilled, then lim Jf dJ1(l = Jf dJ1 for each bounded continuous f: X ~ IR, and this property implies (i)-(v) if X is in addition a completely regular space. PROOF. By definition (i) is equivalent with (iv), and the equivalence of (ii) and (iii) (resp. (iv) and (v)) is immediate. "(ii) => (iv)" Let f: X -+ IR be lower semicontinuous and assume without restriction 0 ~ f ~ 1. From the obvious inequality

1)+ ~ -I L n

(f - -

n

-

l

1{!>i/n}

ni=l

~

f

we get

lim inf

f

f

d~a ~ ~ ~ lim inf ~a({f > ~}) ~ ~ ~ ~({f > ~}). J

where the last expression converges to f dJ1 as n tends to 00. The implication "(iv) => (ii)" is obvious. If f: X ~ lR is bounded and continuous and J1(l ~ J1 weakly, then by (iv) and (v) f dJ1(l ~ f dJ1. Now suppose that X is completely regular and lim f dJ1(l = f dJ1 for all continuous bounded f: X ~ IR. To show (ii) let G ~ X be open and let K £; G be compact. As an immediate consequence of the very definition of complete regularity we fi~d a continuous function f: X ~ [0,1] such thatflK == 1 andflG c == o. Then

J

J

J

J

~a 0 be given. There exist compact sets K 1 ~ B, K 2 ~ BC such that J.l1 (B\K 1) < e and J.l2(B C\K2) < e. A simple compactness argument shows the existence of two open sets G l' G 2 ~ X separating K 1 and K 2 , i.e. K 1 ~ G 1, K 2 ~ G2 and G 1 n G 2 = 0. Then

K 1 ~ G1 ~ G1 ~ G~ ~ K~ and hence G ~

J.l1(B) This holds for all

G

J.l1(K 1) ~ J.l2(K~) ~ J.l2(B)

> 0 so that J.l1 (B)

~

+ e. D

J.l2(B).

In §1 it was shown that given two Radon measures J.l, v on spaces X and ~ there is a unique product Radon measure J.l ® v on X x Ycharacterized by

J.l ® v(C x D) = J.l(C)v(D)

for all compact

C

~

X,

D

~

Y,

and giving for all measurable rectangles the" right" value. In particular the product of two finite Radon measures is finite again (as it should be), and it is only natural to guess that J.l ® v depends continuously (in the weak topology) on both of its arguments.

3.3. Theorem. Let X and Y be two Hausdorff spaces. Then the mapping (J.l, v) ~ J.l ® v from M~(X) x M~(Y) to M~(X x Y) is weakly continuous. PROOF.

In a first step we show that the mapping

X x

M~(Y) ~ M~(X x

Y)

(x, v)~Gx ® v

is continuous. Assume that XiX ~ x and Vix ~ v. Let G 1 , ••• , Gn ~ X and H l' ... , H n ~ Y be open and put U := 1 (G i x Hi). We show first that

u?=

lim inf e xex ® vix(U) ~ This holds trivially if x ~

u?=

1

® v(U).

ex

G i • Suppose now that

1:= {i ~ nix Then there exists some ~o such that we get

E

GJ

+0.

XiX E niEI

G i for all ~ ~ ~o and for those

~

hence lim inf Gxex ® vix(U)

~ lim inf vix(.U Hi) lEI

~ v(.U Hi) lEI

=

ex

® v(U).

48

2. Radon Measures and Integral Representations

Every open set U ~ X x Y has the form U = U;'EA (G;. x H;.) for suitable open sets G;. ~ X and H;. ~ Y. By Theorem 1.5 we can find, given G > 0, finitely many At, ... , An E A such that

ex ®

V(Vl(G A, x RAJ) > ex ® v(U) -

e.

This implies lim inf ex « Q9 v,,(U)

~ lim inf ex

«

Q9

V{Vl (G A, x RAJ)

~ ex Q9 V(Vl (G A, x RAJ) > ex Q9 v(U) -

e.

Hence lim inf Gx ", (8) v(X(U) ~ Gx ® v(U) and 3.1 implies Gx ", ® V(X ~ Gx ® v. The second step will now be an easy consequence of the first one. Let f: X x Y ~ [0, 00] be lower semicontinuous. From the continuity of (x, v) ........ Gx ® v we get that

X x

M~(Y) ~

(x, v) 1-+

[0, 00]

r f(x, y) dv(y) = sup Jyr (n

Jy

1\

f(x, y» dv(y)

n

is also lower semicontinuous, and using the fact that M~(Y) is again a Hausdorff space as well as the Fubini theorem for lower semicontinuous functions (1.12), we may repeat this argument and conclude that M~(X) x M~(Y) ~

(Ji, v) ........

[0, 00],

fxJr

f(x, y) dv(y) dJi(x) =

y

f

f d(Ji (8) v)

Xxy

is lower semicontinuous, too. If now f: X x Y ~ IR is lower semicontinuous and bounded, then f + c ~ for some C E IR, and

°

(j1, v) 1-+

f f d(j1 Q9 v) = f (f + c) d(j1 Q9 v) -

Cj1(X)v(Y)

is again lower semicontinuous, the second term on the right being a continuous function of Ji and v. This finishes our proof. D

3.4. Corollary. If S is a Hausdorff topological semigroup, then so is with respect to convolution.

M~(S)

PROOF. The mapping (Ji, v) ........ Ji * v is continuous as composition of (Ji, v) ........ Ji (8) v and the mapping : M~(S x S) ~ M~(S) defined by (K):= K+, where + denotes the semigroup operation and K+ is the image of K under + ~ cf. 1.15~ 1.16 and Exercise 3.7 below. D

49

§3. Weak Convergence of Finite Radon Measures

It is a general principle in mathematics to approximate complicated functions (or other objects) by simpler ones. Among the Radon measures the so-called molecular measures (i.e. finite positive linear combinations of one-point (or "atomic ") measures) are considered to be "simple" objects. They are dense in the set of all finite Radon measures even in a stronger sense than with respect to the weak topology.

3.5. Proposition. For every Hausdorffspace X the set of molecular measures is a dense subset of M~ (X) with respect to the pointwise convergence on 86(X). PROOF. As the directed set we choose the family A of all finite Borel partitions of X, i.e. the set of all ri = {B 1 , ••• , B n } ~ 86(X) such that B i =F 0, B i n Bj = 0 for i =F j and U7= 1 B i = X, ordered by refinement. For such ri we put, given any f.1 E M~(X), f.1rJ.:= 2:7=1 f.1(BJe xi , where Xi E B i is chosen arbitrarily. Now let B E 86(X)\ {0, X} be given; then for all ri E A finer than rio = {B, Be} we have f.1rJ.(B) = f.1(B); and f.1rJ.(B) = f.1(B) for all ri E A when BE {0, X}. D 3.6. Exercise. Let (f.1rJ.) and f.1 be finite Radon measures on the Hausdorff space X and let them all be concentrated on the Borel subset Y ~ X. Then f.1rJ. -+ f.1 weakly in M~ (X) if and only if f.1rJ.1 Y -+ f.11 Y weakly in M~ (Y). 3.7. Exercise. Let X and Y be two Hausdorff spaces and letf: X -+ Y be a continuous mapping. Then for any f.1 E M~(X) the image measure f.1 f belongs to M~(Y), cf. Proposition 1.15. Show that the transformation f.1 ~ f.1 f from M~(X) to M~(Y) is continuous. 3.8. Exercise. Let X be a Hausdorff space and M~(X) the set of Radon probability measures on X, i.e. M~(X) = {f.1 E M~(X)If.1(X) = I}. Let E := {ex IX E X} be the set of all one-point measures. Show that every {O, 1}valued measure f.1 E M~(X) already belongs to E. Show further that E is a weakly closed subset of M~(X) homeomorphic to X and that E = ex(M~(X». (For the notion of an extreme point see 5.1.) 3.9. Exercise. Let f.1 be a Radon measure on the Hausdorff space X. For K E %(X) define f.1K(B):= f.1(B n K) and f.1K(B):= f.1(B n K)/f.1(K) (if f.1(K) > 0). Show that the net (f.1K) converges to f.1 pointwise on 86(X) and that

li~

f

g d,uK =

f

g d,u

for every f.1-integrable function g: X -+ C. Show that if f.1(X) < (f.1K) (resp. (f.1K» converges weakly to f.1 (resp. f.1/f.1(X».

00

then

3.10. Exercise. Let (prJ.) denote a net of probability Radon measures on the product X x Y of two Hausdorff spaces, and denote by JlrJ. (resp. vrJ.) the

50

2. Radon Measures and Integral Representations

marginal distribution of Pa. on X (resp. Y) (i.e. the image measures under the two canonical projections). If (/la.) converges to /l and ( va.) converges to some one-point measure ey, then (Pa.) tends to /l (8) eye

3.11. Exercise. Let (S, +, 0) denote a Hausdorff topological abelian semigroup with neutral element 0 (cf. 1.16) and let Ji E M~(S). Then 00

Ji*n

n=O

n!

L- =

lim (eo

+ Ji/n)*n

n-+oo

holds in the sense that both limits exist and agree for all Borel subsets of S. Their common value is often called the exponential of Ji and abbreviated exp(Ji). In particular both limits exist and agree with respect to the weak topology.

3.12. Exercise. Let Sand T be two Hausdorff topological semigroups and let h: S ~ T be a continuous homomorphism. Then for Ji, v E M~(S) we have

§4. Vague Convergence of Radon Measures on Locally Compact Spaces In this section X denotes a locally compact Hausdorff space. The vector space CC(X) of continuous functions j: X ~ IR with compact support and the vector space M(X) of signed Radon measures on X form a dual pair under the bilinear form Ji

E

M(X),

4.1. Definition. The vague topology on M(X) is the weak topology a(M(X), CC(X», i.e. the coarsest topology in which the mappings /l ~ (Ji,j> are continuous, when j ranges over CC(X), cf. 1.3.10. In particular the vague topology is a Hausdorff topology. We first remark that for any lower semicontinuous functionj: X the function J11---+

~

[0,00]

f f dJ1

is lower semicontinuous on M + (X) with the vague topology. In fact, from Urysohn's lemma it follows that any suchjis the supremum of the upward filtering family of functions qJ E CC(X) satisfying 0 ~ qJ ~ j,

§4. Vague Convergence of Radon Measures on Locally Compact Spaces

51

so by Theorem 1.5

f f dll sup{f cp dill cp =

E

CC(X), 0

~ cp ~ f}-

A combination of this remark and Theorem 3.1 immediately gives the following relationship between the weak topology on M~(X) and the restriction of the vague topology to M~ (X):

4.2. Proposition. Let (/l(l)(l E A be a net on M ~ (X) and let /l E M ~ (X). Then (/l(l) converges weakly to /l if and only if (/l(l) converges vaguely to /l and lim /l(l(X) = /leX). 4.3. Corollary. The vague topology and the weak topology coincide on the set of Radon probability measures.

M~(X)

Under some extra assumptions on a net (/l(l)(lEA in M +(X) the vague convergence implies the convergence of f d/l(l)(lEA for certain functions f E C(X)\CC(X). As an important example we have

(J

4.4. Proposition. Let (/l(l)(lEA be a net on /l E M~(X). If C :=

then /leX) ~ c, andfor allf

E

M~ (X)

sup /l(l(X)
0 there exists

qJ E

CC(X) such that

qJ II 00 ~ E, and therefore we find

and now it is easy to see that (1) holds.

D

The following result characterizes the relatively compact subsets M ~ M +(X) in the vague topology, i.e. the subsets M for which the vague closure M is vaguely compact.

52

2. Radon Measures and Integral Representations

~

4.5. Theorem. A subset M only if

M +(X) is relatively vaguely compact if and

sup {III EM}


O. The set {II

E M~(X)IIl(X) ~

c}

is vaguely compact. PROOF. The set in question is closed by Proposition 4.4 and relatively compact by Theorem 4.5. D

4.7. Corollary. Suppose X is a compact space. Then weak (or vague) topology.

M~(X)

is compact in the

PROOF. By Proposition 4.6 we have that M~(X) is relatively compact in the vague topology, but also closed since 1 E CC(X). The proof is finished by the observation in Corollary 4.3. D The following result was established by Choquet (1962) in his treatment of the moment problem.

4.8. Proposition. Let V be an adapted space of continuous .functions on a locally compact space X and let L: V -+ IR be a positive linear functional. The set of representing measures, i.e. the set for all

f

E

V}

is convex and compact in the vague topology. PROOF. It is clear that C is convex. For qJ E CC+(X) there existsf that qJ ~ f (cf. the proof of Theorem 2.7), hence



;£

ff

dll = LU)

for

11

E

C,

E

V+ such

§4. Vague Convergence of Radon Measures on Locally Compact Spaces

53

so C is relatively compact by Theorem 4.5. Let (1l(l)(lEA be a net from C converging vaguely to 11 E M + (X) and let.f E V+ . By the remark following 4.1 we get

If

I

~ lim inf f

dll

dll a

= L(f).

We choose g E V+ such thatf E o(g), and let e > 0 be given. There exists a compact set K ~ X such thatf(x) ~ eg(x) for x E X\K, and if qJ E C~(X) is such that 1K ~ qJ ~ 1 we havef(1 - qJ) ~ ego From

I

f d(1l - Ila) =

I

f(1 - cp) d(1l - Ila)

+

I

fcp d(1l - Ila)

we therefore get

and sincefqJ

E

CC(X) the last term tends to zero, so we have

IIf Since

f

E

f,

dll - L(f)1

~ 2t:L(g).

> 0 is arbitrary we get Jf dll = L(f) for all.f E V+ and then for all 'I

D

V, thus proving that 11 E C.

Let 11 be a signed Radon measure on X and let G ~ X be an open subset. Then G is itself locally compact and if.f E CC(G) is extended to X by f(x) = {f(X),

0,

X E

G,

x E X\G,

then .f E CC(X). The mapping f 1--+ Jf dll is a continuous linear functional on CC(G), hence by Theorem 2.5 represented by a signed Radon measure on G, denoted III G, and called the restriction of 11 to G. In case of a (nonnegative) Radon measure 11 on X the restriction III G is ofcourse the usual set-theoretical restriction, introduced already after Remark 1.6. Vague convergence of Radon measures is a local concept as the following result shows: 4.9. Theorem. Let (G(l)(lED be an open covering of a locally compact space X. A net (lli)iEI of signed Radon measures on X converges vaguely to 11 E M(X) if and only ~f (Ili IGCX)iE I converges vaguely to (Ill G(l) for each tYw E D. PROOF. The" only if" part is obvious, so suppose that (Ili IG(l)i E I converges to (Ill G(l) for each tYw E D. Letf E CC(X) have the compact support C, and choose tYw 1 , . . . , tYw n ED such that C ~ G(ll U G(l2 U ... u G(ln. By Lemma 1.17 there exist compact sets C k ~ GCXk k = 1, ... , n such that C = C 1 U ... U Cn. By Urysohn's lemma there exist functions qJk E CC(X) such that 1Ck ~ qJk ~ 1Gock

54

2. Radon Measures and Integral Representations

and SUPP(qJk) ~ G(lk' k

= 1, ... , n. The functions f(x)

!k(x)

nqJk(X)

,

XE

C,

X E

X\C,

L qJj(x)

:=

j= 1

0,

belong to CC(X), sUPP(h) ~ G(lk andf

= Lk= 1 fk' so we have for

and the assertion follows.

i E I,

D

The following result will be used occasionally. For the proof, see e.g. Bauer (1978, p. 233).

4.10. Proposition. Suppose that the locally compact space X has a countable base for the topology. Then the vague topology on M + (X) is metrizable. 4.11. Exercise. Let X be locally compact. For a net (Ji(l) on M +(X) and Ji E M + (X) the following conditions are equivalent: (i) Ji(l ~ Ji vaguely; (ii) lim sup Ji(l(K) ~ Ji(K) for each compact K ~ X and lim inf Ji(l(G) ~ Ji(G) for all relatively compact open sets G ~ X; (iii) lim Ji(l(B) = Ji(B) for all relatively compact Borel sets B ~ X such that Ji(oB) = O.

4.12. Exercise. Show that the set of all Radon measures on a locally compact space taking only values in No = {O, 1, 2, ... , oo} is vaguely closed. 4.13. Exercise. Let the Radon measures Ji(l tend vaguely to Ji and assume that all the Ji(l are concentrated on the closed subset Y ~ X. Then Ji is concentrated on Y, too, and Ji(l tends vaguely to Ji also on the locally compact space Y. 4.14. Exercise. Show that (Ji, v) 1---+ Ji ® v is vaguely continuous as a mapping from M +(X) x M +(Y) to M +(X x Y) for two locally compact spaces X and Y. Hint: Use the Stone-Weierstrass theorem. 4.15. Exercise. Property (ii) in Exercise 4.11 above makes sense on any Hausdorff space X and hence can be used to define vague convergence of Radon measures in this generality. Show however that if M +(X) with respect to vague convergence is a Hausdorff space, then X is necessarily locally compact. Hint: If X o E X has no relatively compact neighbourhood, then for a certain net (X(l) in X we have X(l ~ XO , GXat ~ GXQ and GXat ~ o.

§5. Introduction to the Theory of Integral Representations

55

§5. Introduction to the Theory of Integral Representations Throughout this section E denotes a locally convex Hausdorff topological vector space over IR. The theory below can be applied to complex vector spaces by restricting the multiplication with scalars to real scalars. The basic observation, which should be kept in mind when reading the following general theorems, is that a convex polyhedron is the convex hull of its corners, or equivalently, that any point in the convex polyhedron is the centre of gravity of a molecular probability measure on the corners. The basic notion is that of an extreme point of a set, which is a generalization of a corner of a convex polyhedron and defined below. We shall limit ourselves here to those parts of the theory of integral representations which will be needed in the sequel. A detailed exposition can be found in Alfsen (1971) or Phelps (1966).

5.1. Definition. Let A ~ B ~ E be subsets of E. Then A is called an extreme subset of B if for all x, y E B and A E ]0, 1[: AX

+ (1

- A)Y

E

A ~ x, YEA.

A point a E A is called an extreme point of A if {a} is an extreme subset of A. It is easy to see that if A is convex then a only if for all x, yEA: a = !(X

+ y) ~ x

E

A is an extreme point if and

= y = a.

The set of extreme points of A is denoted ex(A) and in some literature called the extreme boundary of A. Notice that" extreme subset of" is a transitive relation in the set of subsets of E. The above definitions of course make sense for an arbitrary real vector space without topology. Let Xl' .•• , X n be points in E and let ,1.1' ••. , An be numbers E [0, 1] with with 1 Ai = 1. The corresponding convex combination of Xl' ..• , X n is the point

L?=

n

b

=

L Ai x i·

i= 1

If J.1 denotes the molecular measure on E with mass Ai at Xi' i.e. n

J.1

=

L AiGxi'

i= 1

56

2. Radon Measures and Integral Representations

then the point b is called the barycentre of JL For any (continuous) linear functional.f on E we have

f(b) =

Jl

AJ(X i ) =

f

f dJ1,

and this is the motivation for the following:

5.2. Definition. Let X be a compact subset of E and let J1 E M~(X) be a Radon probability measure on X. A point bEE is called the barycentre of J1 if and only if

f(b) =

Ix f dJ1

for all j'

E

E'.

Remark. There exists at most one barycentre of J1 E M~(X). In fact, if b l , b 2 are barycentres of J1 we have .f(b l ) = j'(b 2 ) for all.f E E', and since E' separates the points of E we find b l = b2 . As well as barycentre one also encounters the words centre o.f gravity and resultant. The barycentre is "the value of" the vector integral

Ix x dJ1(x), but such a vector integral need not always converge in E, so a barycentre need not exist. We have, however, the following result:

5.3. Proposition. Let X be a compact subset of E such that K = conv(X) is compact. Then.for every J1 E M~(X) the barycentre exists and belongs to K. Conversely, every point x E K is barycentre of some J1 E M ~ (X). PROOF.

For fEE' and J1

E M~(X)

HI =

{x

E

we put

Elf(x) =

ff

dJ1}

The intersection of the H f's for fEE' is the set of barycentres of J1, hence either empty or a singleton. We shall show

nH

feE'

f

n K =t=

0·

The set H f n K is a closed subset of K, so by a result from general topology it suffices to prove that (1)

for an arbitrary finite subset {ft, ... ,fn} ~ E'. For such a subset we define a continuous linear mapping T: E ~ lR n by

T(x) = Cft(x), ... ,fn(x»,

~5.

57

Introduction to the Theory of Integral Representations

and claim that p:=

(5

fl djl, ... ,

5

fn djl)

E

T(K),

which shows (1). If the contrary is true, there exists by the separation theorem 1.2.3 a linear form qJ: IR n ~ IR such that sup qJ(T(K)) < qJ(p).

(2)

The linear form qJ is given as qJ(x) = for some a and defining g = = 1 a i /; E E', (2) can be expressed

I?

= (at, . .. ,an) E ~n,

sup g(K) < { g djl, which is impossible, J1 being a probability. We next have to prove that any x E conv(X) is the barycentre of some J1 E M~(X). This is clear if x E conv(X), in fact such a point is the barycentre of a molecular measure as remarked earlier. For x E conv(X) there exist nets (x~)~ E A of points from con v(X) con verging to x and (J1~)~ E A of molecular measures from M~(X) such that x~ is the barycentre of J1~ for each ~ E A. By Corollary 4.7 there exist J1 E M~(X) and a subnet (J1~p) converging weakly to J1. For fEE' we then have

5

f djl

= li~

5

f djlap

= li~

f(x ap )

which shows that x is the barycentre of J1.

= f(x), D

5.4. Remark. If E is complete then conv(X) is compact for every compact X ~ E, so every J1 E M~(X) has a barycentre in this case. This applies in particular to Frechet spaces and Banach spaces. For details, see, e.g. Robertson and Robertson (1964). Already Minkowski proved that a compact convex set K in ~n is the convex hull of ex(K). In 1940 Krein and Milman found a far-reaching generalization of Minkowski's result:

5.5. Theorem. Every compact convex set K in E is the closed convex hull of its extreme points, i.e. K

= conv(ex(K)).

We first show that any nonempty compact set C has extreme points. We form the family ~ of nonempty, closed extreme subsets of C. Notice that C E~. A Zorn's lemma argument shows that ~ contains a minimal element M with respect to inclusion. To see that M has only one point, which is then an extreme point of C, we assume the existence of x, y E M, PROOF.

58

x

2. Radon Measures and Integral Representations

+y, and choosef E E' such thatf(x) > f(y). Then M o = {zEMlf(z) = supf(M)}

is easily seen to be an extreme subset of M, hence MoE ~ Since M 0 is a proper subset of M we are led to a contradiction. Clearly conv(ex(K» is a compact convex subset of K. If they are not equat there exists by the separation theorem 1.2.3 an fEE' such that

supf(K) > supf(conv(ex(K»).

(3)

The set

M = {x

E

Klf(x) = supf(K)}

is a nonempty compact subset of K, and by the first part of the proof ex(M) 0. Since M is an extreme subset of K we have ex(M) ~ ex(K), but this is impossible due to (3). D

+

For every subset A ~ E we have conv(A) = conv(A). An equivalent formulation of the Krein-Milman theorem is therefore that K = conv(ex(K». Using Proposition 5.3 we can reformulate the theorem in the following way:

5.6. Theorem. Let K be a compact convex set in E. Every x centre of a measure Jl E M~(ex(K».

E

K is the bary-

A natural and important question in connection with the above theorem is whether Jl can be chosen such that Jl is concentrated on the set of extreme points, i.e. such that Jl(K\ex(K» = O. The answer is yes if K is metrizable, and this is the content of Choquet's theorem. Tbe answer is no in general if K is nonmetrizable, simply because ex(K) can be a nonmeasurable subset of K in this case. There is however a satisfactory solution to the question in the nonmetrizable case also, related to the notio.n of a boundary measure. For a compact convex subset K of E we define a partial ordering <J on M~(K) by Jl 0 sufficiently smalt we get two matrices A+ and A_ in On such that A = t(A+ + A_).

61

Notes and Remarks

5.16. Exercise. Let K be the unit ball {x E Hlllxli ~ I} ofa Hilbert space H. Show that ex(K) is the topological boundary of K, i.e.

ex(K)

= {x E HI IIxll = I}.

5.17. Exercise. Let (X, d) be a measurable space and let E be the Banach space over [K of bounded .91 -measurable functions f: X -+- [K with the uniform norm IIf'II = sup{lf(x)llx E X}. Show that the set of extreme points of the unit ball {f E E III f I ~ I} of E is the set of functionsf E E with If(x)1 = 1 for all x EX. Formulate and prove a similar result when X is a compact Hausdorff space and E = C(X). 5.18. Exercise. Let K denote the set of con vex functions f: ]0, 2IJ [ -+- [0, 1] considered as a subset of IRl O, oo[ with the topology of pointwise convergence. Show that K is a metrizable compact convex set. Show that every f E K is decreasing, continuous and differentiable from the left and the right. For E K and t > 0 we define

.r

!rex) = {f(t) + (x - t)f'-(t) f(x)

for 0 < x ~ t, for x > t.

Show that!r, f - !r E K and deduce that the extreme points of K are the following functions qJo == 0, qJoo == 1, qJt(x) = (1 - x/t) + , 0 < t < 00. Show finally that K is a Bauer simplex. (This way of finding the extreme points of K is taken from Johansen (1967).) 5.19. Exercise. Show that the Riesz representation theorem on a compact Hausdorff space is a special case of the Krein-Milman theorem. 5.20. Exercise. (Douglas 1964). Let V be an adapted space of continuous functions on a locally compact space X and let C be the convex set of representing measures for a positive linear functional L: V -+- IR. Show that J1 E C is an extreme point of C if and only if V is dense in 2 1(X, J1). 5.21. Exercise. Let K and L be compact convex subsets of locally convex spaces~ and letf: K -+- Lbe continuous~ affine and onto. Show that ex(L) ~ f(ex(K))~ and use this result to determine the set of extreme points of

L:= {( ft df.1(t), ft df.1(t),···, ft" df.1(t)) If.1 E 2

M~([O, I])}.

Notes and Remarks We have not assumed a Radon measure to be locally finite. The reason for this is simply that we do not need this condition to derive any of the main results and, on the other hand, on many spaces a Radon measure is automatically locally finite. Certainly this is the case for locally compact spaces

62

2, Radon Measures and Integral Representations

but it holds, for example, also for metric spaces: Let fl be a Radon measure on the metric space X and suppose there is some x E X such that fl(G) = 00 for each open set G containing x, then in particular fl(B n ) = 00 if B n is the open ball of radius lin around x, and hence fl(K n ) ~ n for a suitable compact set K n ~ B n , where without restriction x E K n • Now the point is that K:= U:= 1 K n is again compact, because if K ~ UAEA GA for a family of open sets (G A)' then for some Ao we have x E GAo' and then for some no x E B no ~ GAo implying U:=no K n ~ GAO' The fact that fl(K) = 00 shows that our assumption was wrong. Exercise 1.30 shows that non locally finite Radon measures may occur, and this depends on the fact that each compact subset is finite. As another example where this is the case we mention the fine topology of potential theory, for instance on [R3, cf. Helms (1969). The fine topology is by definition the coarsest topology on 1R 3 in which all superharmonic functions are continuous. The fine topology is completely regular (Brelot 1971, p. 5), and the fact that every finely compact set is finite is proved in Helms (1969,

p.208). Next we want to relate our approach to Radon measures with the "classical" one as developed, for example, in Bourbaki (1965-1969). There, as already mentioned in the introduction, the "functional point of view" is prevalent, a (Radon) measure fl being by definition a positive linear form on CC(X), if X is locally compact. Two set functions are then considered, fa mesure exterieure fl* defined by fl*(G) = sup{l.f

E

CC(X), 0 ~

f

~

IG}

for open

G

~

X

and /l*(A)

= inf{fl*(G)IA

~

G, G open}

for

A ~ X,

and fl' derived from l'integrale superieure essentielle and given by /l'(A)

=

SUP{fl*(A n K) IKE %(X)}

for

A ~ X.

Both fl* and fl' are Borel measures, i.e. a-additive when restricted to 8l(X), and fl' is a Radon measure in our sense whereas /l* is not so in general. One has fl' ~ fl* and they agree on open sets and on Borel sets B with /l*(B) < 00, and in particular on compact sets. It follows that for locally compact and a-compact spaces one has /l'(B) = /l*(B) for all BE 8l(X), so for these spaces Bourbaki's notion of a Radon measure is equivalent to ours. This holds in particular for compact spaces. Bourbaki (1965-1969, Ch. IV, §L Ex. 5) gives an example where fl'(r') = 0 and fl*(r') = 00 for a certain closed subset r' in some locally compact space, and this shows that fl* is not a Radon measure in our sense. Bourbaki defines a (Radon) premeasure on a Hausdorff space X as a mapping W which to every compact subset K ~ X associates a Radon measure WK on K such that WK IL = WL if L is a compact subset of K. Then the following set function W' is considered W'(A) = sup{(WK)'(A n K)IK

E

%(X)}

for

A

~

X.

63

Notes and Remarks

The restriction of W· to f4(X) is a Radon measure in our sense. Bourbaki calls W a (Radon) measure if W· is in addition locally finite. It follows that Bourbaki's notion of (Radon) measures is equivalent with locally finite Radon measures in our sense. Let X be a Hausdorff space and consider Borel measures v: f4(X) --+ [0, 00 ] satisfying (a) (b) (c)

v(K) < 00 for K E f(X); v(G) = sup{v(K)IK s; G,KEf(X)} foropen G s; X; v(B) = inf{ v(G) I B s; G, G open} for B E fJ6(X).

There is a one-to-one correspondence between locally finite Radon measures J.l on X and Borel measures v satisfying (a), (b) and (c). In fact, if J.l is a locally finite Radon measure then J.l*(B) = inf{J.l(G)IB s; G, G open}

for

B

E

fJ6(X)

is a Borel measure satisfying (a), (b) and (c), and if v has these properties then v·(B)

= sup{ v(K) I K

s; B, K E f(X)}

for

B

E

fJ6( X)

is a locally finite Radon measure. Furthermore (J.l*r = J.l and (v·)* = v. For the proof of these assertions see Schwartz (1973), where a third equivalent definition of a locally finite Radon measure is given, namely as a pair (m, M) of Borel measures satisfying certain conditions realized by (J.l, J.l*) and (v·, v) in the above notation. Notice that a Borel measure v satisfying (b) and (c) is locally finite if and only if (a) holds. In the locally compact case the exterior measure J.l* satisfies (a), (b) and (c) and (J.l*r = J.l., (J.l.)* = J.l*. The generalized monotone convergence theorem (1.5) involves only the values of the underlying Radon measure J.l on open sets, so it holds for any Borel measure v which agrees with J.l on open sets. In particular, we have

°

for each lower semicontinuous function f ~ on X. For a continuous real-valued functionj'integrability with respect to J.l and v are equivalent and Sf dJ.l = Sf dv in case of integrability. Bourbaki (Ch. IX, §3) also considers the possibility of "extending" a set function A: f(X) --+ [0, oo[ to a Radon measure, however the crucial property (1) of our §1, the defining property of a Radon content, which goes back to Kisynski (1968), is not discussed there. Theoreme 1 of §3 in Bourbaki should be compared with our Lemma 1.3. Theorem 1.4 is due to Kisynski. Replacing the Hausdorff space X by an abstract set and the family f(X) of compact subsets of X by a suitable set system called "compact paving", Tops¢>e (1978) proved an abstract measure extension theorem which not only contains Theorem 1.4 but also, for example, Caratheodory's classical result.

64

2. Radon Measures and Integral Representations

It should be mentioned that on many "nice" spaces a finite Borel measure is automatically a Radon measure. This holds in particular on Polish spaces, i.e. separable and completely metrizable spaces, cf. for example, Bauer (1978, Satz 41.3), but it can even be shown for so-called analytic spaces, i.e. Hausdorff spaces which are the continuous image of some Polish space; see Dellacherie and Meyer (1978, Chap. III) for a proof. The last-mentioned book also contains a proof of the bimeasure theorem for finite Radon bimeasures on separable metric spaces. A slightly more general version may be found in Morando (1969), but both results are in fact a special case of a theorem of Marczewski and Ryll-Nardzewski (1953) about nondirect products of measures. The Riesz representation theorem is certainly a cornerstone of functional analysis. Our proof is based on Pollard and Tops¢e (1975) and we refer to the references given there, in particular to Batt (1973) for further information on this important topic. The theory of adapted spaces has had important applications in potential theory, see Sibony (1967-1968). The theory of weak convergence of finite (or probability) Radon measures is mainly motivated by its applications in probability theory and mathematical statistics. For a thorough treatment on metric spaces we refer to Billingsley (1968) and Parthasarathy (1967). Later on Tops¢>e (1970) discovered that a satisfactory theory of weak convergence can be developed on arbitrary Hausdorff spaces. For a completely regular space X the weak topology on M~(X) is induced by the weak topology a(Mb(X), Cb(X)). However, for Hausdorff spaces in general it is not possible to extend the weak topology from M~(X) to Mb(X) in such a way that Mb(X) is a Hausdorff topological vector space. In fact, if such an extension was possible then Mb(X) would be completely regular and so would {£xlx EX}, which is homeomorphic to X by Exercise 3.8. A particularly important topic for probabilistic applications is the characterization of relatively compact subsets of M~(X) in the weak topology, and we should mention the striking result due to Prohorov: For Polish spaces X a subset M ~ M~(X) is weakly relatively compact if and only if for each £ > 0 there is a compact set K ~ X such that SUPjiEM J1(X\K) < £, a condition on M called uniform tightness, see Billingsley (1968, Theorems 6.1 and 6.2). Theorem 3.3 may be found in Ressel (1977); Exercise 3.10 is a generalization of Slutsky's theorem, cf. Ressel (1982b). For a locally compact space X the space CC(X) is often equipped with the inductive limit topology of the Banach spaces CK(X), K E f(X), appearing before Theorem 2.5. With this topology, which is equal to the topology gi ven in Exercise 2.11 by the family 14 , CC( X) is a barrelled space, and the topological dual space is M(X) with the vague topology. This approach is the starting point in Bourbaki (1965-1969), who also seems to be the first who has systematically studied the vague topology. Theorem 4.5 is a special case of the Alaoglu- Bourbaki theorem, cf. Exercise 1.3.11. In the special case of X = IR Theorem 4.5 is sometimes called Helly's selection theorem. In

Notes and Remarks

65

fact it follows from 4.5 and 4.10 that any sequence of measures in M~(IR) with bounded total mass has a vaguely convergent subsequence. The result in Exercise 4.15 is due to TopscPe. The importance of the theory of integral representations lies undoubtedly in the fact that it gives a unified approach to a great number of classical formulas and theorems, cf. Phelps (1966). Let usjust mention here Herglotz' formula for nonnegative harmonic functions in a ball, the far more general Martin representation, and the theorems of Bernstein and Bochner. In Chapter 4 we shall use the theory to prove integral representation theorems for positive definite functions on abelian semigroups. The idea of considering a point in a metrizable compact convex set K as the barycentre of a probability measure concentrated on ex(K) is due to Choquet, and the whole theory is often called Choquet theory. In our applications of the theory we use only the special case where ex(K) is closed, in which case the representation theorem is equivalent with the Krein-Milman theorem. Therefore we have given a complete proof of the latter and only indicated the general results, which can be found in many books, for example, Alfsen (1971) and Phelps (1966).

CHAPTER 3

General Results on Positive and Negative Definite Matrices and Kernels

§1. Definitions and Some Simple Properties of

Positive and Negative Definite Kernels When dealing with positive and negative definite kernels a certain amount of confusion often arises concerning terminology. A positive definite kernel defined on a finite set is usually called a positive semidefinite matrix. Sometimes it is only called "positive", which may be misleading. When working on groups, the name positive definite function is used traditionally. In our previous papers on abelian semigroups we also followed this tradition. Instead of calling a kernel t/J negative definite, some authors call the kernel - t/J "conditionally positive definite" or "almost positive." In this book we use mainly the larger class of "semidefinite" kernels of all kinds and therefore prefer to avoid the prefix" semi" which otherwise would appear several hundred times. Adapting the above point of view, an n x n matrix A = (ajk) of complex numbers is called positive definite if and only if n

L

cjcka jk

~ 0

j,k= 1

for all {c 1 , ••• , Cn} ~ c. It is well known that this is the case if and only if A is hermitian (i.e. ajk = akj for j, k = 1, ... , n) and the eigenvalues of A are all ~ o. Similarly A is called negative definite if and only if A is hermitian and n

L

j,k= 1

cjckajk

~0

§I. Definitions and Some Simple Properties of Positive and Negative Definite Kernels

67

for all {c l , ... , Cn} ~ C with the extra condition Lj= I Cj = O. (This definition requires n ~ 2. Any 1 x 1 matrix A = (all) with real all is called negative definite.)

1.1. Definition. Let X be a nonempty set. A function qJ: X x X called a positive definite kernel if and only if

--+

C is

n

L

j,k= I

CjCkqJ(Xj, Xk) ~ 0

for all n E N, {Xl' ... ' X n } ~ X and {C l , ... , Cn} ~ C. We call the function qJ a negative definite kernel if and only if it is hermitian (i.e. qJ(Y, x) = qJ(x, y) for all x, Y E X) and n

L

j,k= I

CjCkqJ(xj, Xk) ~ 0

for all n ~ 2, {Xl' ... , x n} ~ X and {c l , ... , cn} ~ C with Lj= I Cj = O. If the above inequalities are strict whenever X I' ... 'Xn are different and at least one of the C I' ... , Cn does not vanish, then the kernel qJ is called strictly positive (resp. strictly negative) definite.

1.2. Remark. In the above definitions it is enough to consider mutually different elements Xl' ... , Xn E X. In fact, if Xl' ... , Xn E X are arbitrary and x a1 ' • • • , x ap are the mutually different elements among the x/s, then n

L

p

j,k= I

cjCkqJ(x j , Xk) =

L

j,k= I

djdkqJ(x aj , x ak ),

where dk

:=

L

Ci ,

k = 1, ... ,p.

{iIXi=X czk }

Furthermore, if (1: X --+ X is a bijection, then qJ is a positive (resp. negative) definite kernel if and only if qJ «(1 x (1) is a positive (resp. negative) definite kernel. If X is a finite set, say X = {Xl' ... , x n }, then plainly qJ is positive (resp. negative) definite if and only if the n x n matrix 0

(qJ(Xj' Xk»l ~j,k~n is positive (resp. negative) definite. We now list some simple properties and examples of positive and negative definite kernels.

1.3. A kernel qJ on X x X is positive (resp. negative) definite if and only if for every finite subset X 0 ~ X the restriction of qJ to X 0 x X 0 is positive (resp. negative) definite.

68

3. General Results on Positive and Negative Definite Matrices and Kernels

1.4. If qJ is positive definite, then qJ(x, x) ~ 0 for all x on the diagonal L\:= {(x, x)1 x EX}. 1.5. Let (:

E

X, i.e. qJ is nonnegative

:) be a positive definite 2 x 2 matrix. Then

o~

(1,1)(: :)C) =

a

+b+c+d

= - 1m c. Further,

implying 1m b

o~

0,0(: :)( _~) =

implying Re b = Re c, i.e. b definite kernel is hermitian.

= c.

a - ib

+ ic + d

It follows immediately that any positive

qJ on X x X is positive (resp. negative) definite if and only if qJ is symmetric (i.e. qJ(x, y) = qJ(Y, x) for all x, Y E X) and

1.6. A real-valued kernel n

L

~ 0

CjCkqJ(X j , x k )

(resp.

~O)

j,k= I

for all n E N, {Xl' ... , X n} £ X and {c l , ... , en} £ IR (resp. addition). For, if Cj = Qj + ib j , Qj and b j being real, then n

L

LJ= 1 Cj

=

0 in

n

Cj0c qJ(Xj' x k )

=

j,k= I

L

(QjQ k

+ bjbk)qJ(x j , x k)

j,k= I n

+i L

(bjQ k -

Qjbk)qJ(X j , X k ),

j, k= I

and the last sum is zero if qJ is symmetric.

1.7. A 2 x 2 matrix (:

~) is negative definite if and only if a, d E IR, b = c

and

and this inequality is equivalent with a

+d

~

2 Re b.

Therefore, we have for any negative definite kernell/J the inequality

l/J(X, x)

+

l/J(y, y) ~ 2 Re l/J(x, y).

69

§l. Definitions and Some Simple Properties of Positive and Negative Definite Kernels

1.8. Let (:

(w,

~) be a hermitian 2 x

Z)(: ~)(~) = alwI

2

2 matrix. Then for z,

iC we have

+ 2 Re(bzw) + dlzl 2 2

b 1 a w + ~z

=

WE

2

Izl + -;;(ad -lbI 2 )

(for a =F 0).

I

The matrix is therefore positive definite if and only if a det(:

~) = ad -

Ibl 2

~

0, d

~

0 and

~ O.

Hence for any positive definite kernel qJ we have I

qJ(x, y) 12 ~ qJ(X, x) . qJ(Y, y).

1.9. Iff: X -+ C is an arbitrary function, then qJ(x, y) definite, because

j.~lCjCkqJ(Xj'

Xk)

=

IJl cJ

(X)12

:=

f(x)f(y) is positive

~ O.

The kernel ljJ(x, y) := f(x) + f(y) is negative definite, for if IJ= 1 Cj = 0, then even IJ,k= 1 cjCkljJ(x j , x k) = O. In particular, a constant kernel (x, Y)H C is positive definite if and only if c ~ 0 and negative definite if and only if C E IR. 1.10. The kernelljJ(x, y) = (x - y)2 on IR x IR is negative definite, + Cn = 0 implying (for real numbers Cj' see 1.6)

In j,k=l

CjCk(X j - Xk)2 = -2

(

In

CjX j

C1

+ ...

)2 ~ o.

j=l

1.11. If X is a nonempty set, then the family of all positive (resp. negative) definite kernels on X x X is a convex cone, closed in the topology of pointwise convergence. A very important property of positive definite kernels is their closure under pointwise multiplication which was proved by Schur (1911) (in the case of matrices): 1.12. Theorem. Let qJl' qJ2: X x X -+ C be positive definite kernels. Then qJ2: X x X -+ C is positive definite, too.

qJl .

PROOF. It suffices to prove that if A = (a jk ) and B = (b jk ) are positive definite n x n matrices, then C := (ajkbjk) is positive definite.

70

3. General Results on Positive and Negative Definite Matrices and Kernels

Now it is well known from linear algebra (and also follows from 3.1 below) that there are n functions .fI' ... ,.fn: {I, ... , n} -+ e such that n

ajk = Let

C l' ... , Cn E

L fp(j)fp(k),

for

j, k

p=1

= 1, ... , n.

e be arbitrary, then n

n n cjckajkbjk = L L cjfp(j)ckfp(k)bjk ~ j,k= 1 p= 1 j,k= 1

L

o.

D

1.13. Corollary. Let qJt: X x X -+ e and qJ2: Y x Y -+ e be positive definite kernels. Then their tensor product qJt ® qJ2: (X x Y) x (X x Y) -+ e defined by qJt ® qJ2(X t , Yt, x 2, Y2) = qJt(X t , x 2)· qJ2(Yt, Y2) is also positive definite. PROOF. qJt(X t ,

If (PI: (X x Y) x (X x Y) -+ e is defined by (PI(X t , Yt' X 2, Y2) = x 2) and analogously (P2(X t , Yt, X 2' Y2) = qJ2(Yt, Y2)' then qJt ® qJ2 =

D

(PI . (P2 and is therefore positive definite.

1.14. Corollary. Let qJ: X x X -+ e be positive definite such that IqJ(x, y) I < p for all (x, y) E X X X. Then if f(z) = L:=o anz n is holomorphic in {z E e II z I < p} and an ~ 0 for all n ~ 0, the composed kernel f qJ is again positive definite. In particular if qJ is positive definite, then so is exp( qJ). 0

PROOF. By Theorem 1.12 for each n E ~ the kernel qJn is positive definite, therefore L~= 0 an qJn is positive definite for all N E ~ and so is its pointwise limit f 0 qJ. D

1.15. Remark. In contrast to Theorem 1.12 above the ordinary matrix product of two positive definite matrices is positive definite if and only if the two matrices commute. This follows from the simultaneous diagonalization of these matrices. In particular the matrix exponential of any positive definite matrix again has this property. By using the Jordan decomposition one can show that the matrix exponential of every symmetric real matrix is positive definite (even strictly). The following remarkable criterion for strict positive definiteness is often useful.

1.16. Theorem. Let A strictly positive definite

=

(a jk ) be some hermitian n x n matrix. Then A is

if and only if

det«ajk)j,k~p)

> 0

for p = 1, ... , n. Suppose first A to be strictly positive definite. As in the proof of 1.12 we choose n vectors z t, ... , Zn E en such that

PROOF.

j, k

= 1, ... , n,

§l. Definitions and Some Simple Properties of Positive and Negative Definite Kernels

71

which of course can also be written as A = BB*, where B is the n x n matrix with rows Zl' ..• ' Zn. This implies det A = Idet BI 2 ~ 0 and certainly A cannot be singular so that in fact det A > o. Obviously the same reasoning can be applied to the submatrices (ajk)j,k~P for p = 1, ... , n. For the other direction, we proceed by induction on n. The case n = 1 being trivially true, let us suppose that the theorem holds for n - 1. By assumption all> 0 and we subtract a1k/a11 times the first column from the kth column, k = 2, ... , n. The new matrix (a}k)j,k~n (where the first column remained unchanged whereas for k ~ 2 we have a}k = ajk - (a 1k/a 11 )aj1) has the same principal minors as (ajk), i.e. det«ajk)j,k~p)

=

det«ajk)j,k~p),

p

= 1, ... , n

and if we now change the matrix (a}k) to B, where all

0

0

B=

then still det«bjk)j,k~p) = det«ajk)j,k~p) for all p, and B is furthermore hermitian. Now

for p = 2, 3, ... , n implying by assumption that the (n - 1) x (n - 1) matrix

n

+

L C 1c;a 1 k + k=2

2

I C1 1

all

72

3. General Results on Positive and Negative Definite Matrices and Kernels

If (c 2 , •.. , cn) =t= 0 then the first sum is >0, and for (c 2 , ... , cn) == 0 but CI 0 the second term is strictly positive. Hence A is a strictly positive definite matrix. D

*

One might expect that a corresponding result for positive definite matrices holds if the determinants in the above theorem are only supposed to be nonnegative. The simple counterexample given by the 2 x 2 matrix

(0o 0)

-1 destroys this hope. However, the following result seems to be rather satisfactory, at least theoretically.

1.17. Theorem. Let definite

qJ:

X x X

-+

C be a hermitian kernel. Then qJ is positive

if and only if det«qJ(x j , Xk»j,k~n) ~ 0

for all n

E ~

and all

{Xl' ... ,

~

xn}

X.

PROOF. If qJ is positive definite, then as in the beginning of the proof of the above theorem we see that all determinants in question are nonnegative. Let us on the other hand assume this condition. We define a slightly perturbed kernel qJe:== qJ + e· 1.1' where e > 0 and L\ is the diagonal in X x X. For mutually different elements Xl' ... , X n E X it is easily seen that n

det«qJe(x j , Xk»j,k~n) ==

Ld

p

P e ,

p=o

where dn == 1 and dp ==

L

det«qJ(x j ,

Xk»j,keA)

~ 0

AS{I, ... ,n}

IAI=n-p

for p == 0, 1, ... , n - 1. Therefore det«qJe(x j , Xk»j.k~n) ~ en > 0 implying that qJe is a strictly positive definite kernel. Hence the pointwise limit qJ == lim e _ o qJe is positive definite. D The special case n == 2 has already been derived in 1.8.

1.18. Exercise. If qJ is a positive definite kernel, then also Re qJ, iP and IqJ 12 are positive definite, but not necessarily IqJ I. If t/J is negative definite, then so are Re t/J and If.

1.19. Exercise. For

ZE

C define M z :==

(~ ~ Z

Z

:). Then M z is positive

1

definite if and only if I z I ;£ 1 and [3 - 2 Re(z)] 1 z 12 ;£ 1. For -1 ;£ z < the matrix M z is not positive definite.

-1

73

§2. Relations Between Positive and Negative Definite Kernels

1.20. Exercise. Let H be a complex (pre-) Hilbert space. Then its scalar product > is a positive definite kernel. The squared distance l/J(x, y) := Ilx - YII 2 is negative definite.

O. If exp( - tt/J) is positive definite, then 1 - exp( - tt/J) is, of course, negative definite and so is therefore the pointwise limit

PROOF.

t/J =

lim O O. Instead of proving this we show the following more general result.

t/J: X

2.3. Theorem. Let Jl be a probability measure on the half-line IR+ such that J~ s dJl(s) < 00, and let !RJl denote its Laplace transform, i.e. !RJl(z) = J~ e- sz dJl(s), Z E C+. Then t/J: X x X -+ C+ is negative definite if and only if !RJl(tt/J) is positive definite for all t > O.

o
0 we have !l'j1(tljJ)

=

roo exp( -tsljJ) dj1(s)

pointwise on X x X, which is positive definite, being a mixture of the positive definite kernels exp( - tst/J). If on the other hand !RJl(tt/J) is positive definite for all t > 0, then for each (x, Y) E X X X we get

f.oo

~ [1 _ !l'j1(tljJ(x, Y))] = 1 - exp[ -tsIjJ(x, y)] dj1(s) t o t -+

IjJ(x, y)

roo s dj1(s)

for

t

-+

0,

where we could apply Lebesgue's theorem because of

11 - exp[ -tsljJ(x, Y)]I ~ t

'IjJ(x, y)ls.

-

Being a pointwise limit of negative definite kernels, definite, too.

t/J

itself is negative

0

Choosing Jl = 8 1 in the above theorem, we get back Theorem 2.2 for C+-valued t/J, and the choice of Jl = e- t dt shows, as already mentioned, that t/J: X x X -+ C+ is negative definite if and only if (t + t/J)-l is positive definite for all t > O.

76

3. General Results on Positive and Negative Definite Matrices and Kernels

2.4. Remark. If a probability measure J1 on ~ + has infinite first moment and t/J: X x X -+ C + is negative definite, then 2 J1(tt/J) is still positive definite for all t > 0, but, in general, the converse does not hold. It follows from later results (cf. 2.10 and 4.4.5) that exp( -~) is the Laplace transform of some probability measure J1 on ~+. Now a matrix (a jk ) of the form ajk = (Sj + Sk)2, where S b ..• , Sn ~ 0, is not negative definite in general, C 1 + ... + Cn = 0 cjckajk = 2(I Cj Sj)2, but nevertheless for all t > 0 implying

L

2J1(ta jk )

= exp( -~) = exp( -JiSj) exp( -JiSk)

is positive definite by 1.9. qJ is positive definite and qJ I~ == C for some C E ~ + then obviously C - qJ is negative definite, bounded and vanishes on the diagonal ~. A similar statement in the other direction (which may be found in the literature) is not generally true; see, however, 4.3.15. For Xl = -1, x 2 = 0, X3 = + 1, the 3 x 3 matrix

2.5. Remark. If

is negative definite, vanishes on the diagonal, and is bounded by 4, but for no real number t the matrix (t - ajk) is positive definite, because t

det( t - a jk)

= t- 1 t-4

t -

1

t t-l

t -

4

t -

1 - -8

for all

t E ~.

Negative definite kernels are intimately related to so-called "infinitely divisible" positive definite kernels. 2.6. Definition. A positive definite kernel qJ is called infinitely divisible if for each n E N there exists a positive definite kernel qJn such that qJ = (qJn)n. If t/J is negative definite then qJ = e - '" is infinitely divisible since qJn = exp( -(I/n)t/J) is positive definite and (qJn)n = qJ. Furthermore, qJ has no zeros. Proposition 2.7 below shows, in particular, that every strictly positive infinitely divisible kernel has this form. Let qJ be infinitely divisible. Then

k, n

~

1,

so that the nonnegative kernel IqJ I is infinitely divisible inside the family of all nonnegative positive definite kernels, each (under this restriction uniquely determined) nth root IqJn I again being an infinitely divisible positive definite kernel. Let ~(X) denote the closure of all real-valued negative definite kernels on X x X in the space] - 00, oo]X x x.

~2.

Relations Between Positive and Negative Definite Kernels

77

qJ ~

0 on X x X thefollowing

2.7. Proposition. For a positive definite kernel conditions are equivalent: (i) qJ is infinitely divisible; (ii) -log qJ E ~(X); (iii) qJt is positive definite for all t > O. PROOF. "(i) => (ii)" Let

t/J := -log qJ, then

t/J = lim n[1 - exp( -t/J/n)J E ~(X). "(ii) => (iii)" Let (t/J ex) be a net of (finite) negative definite kernels converging pointwise to t/J = -log qJ. Then for any t > 0 we have exp{ - tt/Jex) -+ exp( - tt/J) = qJt, so that qJt is positive definite by Theorem 2.2. "(iii) => (i)" Take t = !, t, i, ... .

D

2.8. Remarks. (I) The above proof shows that ~(X) is in fact the monotone sequential closure of the subset of all negative definite kernels, bounded above. (2) For

Z E

iC, IZ I ~ 1 the 2 x 2 matrix

G;)

is an infinitely divisible

positive definite kernel with nonuniquely determined positive definite "roots ". We conclude this section by indicating a large class of functions which operate on negative definite kernels. For J1 E M +(]O, oo[) we define g: D(J1) -+ C by

1 00

g(z)

where D{J1) is the set of Z

E

=

(l - e- AZ) df1(A.),

C for which A 1---+ 1 - e-;'z is J1-integrable.

2.9. Proposition. Let t/J: X x X -+ C be a negative definite kernel and let J1 E M +(]O, oo[). Ift/J(X x X) ~ D(J1) then g t/J is negative definite. Furthermore, for X o E X the kernel (x, y) 1---+ g[t/J(x, x o) + t/J(y, xo)J - g[t/J(x, y) + t/J{x o , xo)J is positive definite provided (t/J(X x X) + t/J(x o , x o)) u (t/J(X, x o) + t/J{xo, X)) ~ D(J1). If So A{t + A)-l dJ1{A) < 00 and t/JI~ ~ 0 then g t/J is negative definite and g[t/J(x, x o) + t/J(y, x o)] - g[t/J(x, y)J is positive definite for all X o E X. 0

0

= 1 - e-;'z where A E ]0,00[, X and C t , •.. , Cn E C such that L ci = 0

PROOF. It suffices to prove the result for g(z)

i.e. J1 = we get

G;',

D{J1) = C. If Xl'

L ci Ck{1

i, k

... , X n E

- e-;'t/J(Xj,Xk») = -

L ciCke-;'t/J(Xj,Xk) ~ 0 i, k

as an immediate consequence of Theorem 2.2. For any X o E X the kernel t/J{x, x o) + t/J{y, xo) - t/J(x, y) - t/J(x o , x o) is positive definite by Lemma 2.1,

78

3. General Results on Positive and Negative Definite Matrices and Kernels

as is therefore the kernel exp(A[t/J(X, x o)

+ l/J(y, xo) - t/J(x, y) - t/J(x o , x o)]) -

1

by 1.14, and multiplying with the positive definite kernel exp( - At/J(X, x o)) exp( - At/J(y, x o)) gives exp( - A[t/J(x, y)

+ t/J(xo, xo)]) -

exp( - A[t/J(x, x o)

+ t/J(y, xo)])

= g[l/J(x, x o) + t/J(y, x o)] - g[l/J(x, y) + t/J(x o , x o)], which is positive definite by Theorem 1.12. If J 2

D

3.4. Exercise. Let the kernel qJp be defined on IR+ x ~+ by qJp(x, y) = exp[ -(x + y)P] where 0 < p < 00. Show that qJp is positive definite if and only if p

~

1.

3.5. Exercise. Let H be the RKHS associated with the positive definite kernel qJ: X x X -+ C. Show that any closed subspace of H is the RKHS for some positive definite kernel on X x X.

3.6. Exercise. Let qJ, t/!: X x X

-+ C be two positive definite kernels. Then RKHS(qJ) £ RKHS(t/!) if and only if At/! - qJ is positive definite for some A> 0.

3.7. Exercise. Let qJ: X x X -+ C be a positive definite kernel not identically zero. Show that the following conditions are equivalent: (i) qJ generates an extreme ray in the convex cone 9 of positive definite kernels on X x X (i.e. {AqJ IA ~ o} is an extreme subset of 9). (ii) RKHS( qJ) is of dimension 1. (iii) qJ(x, y) = f(x)f(y) for some functionf: X -+ C not identically zero.

84

3. General Results on Positive and Negative Definite Matrices and Kernels

3.8. Exercise. Let qJ: X x X ~ C be a positive definite kernel such that qJ(x, y) =+= 0 for all (x, y) E X X X. Show that if also l/qJ is positive definite then qJ(x, y) = f(x)f(y) for some functionf: X ~ C\{O}. 3.9. Exercise. Given a positive definite kernel qJ: X x X ~ C and a nonempty subset X' ~ X, consider the restriction qJ' := qJ IX' x X' and the two RKHS's Hand H' associated with qJ (resp. qJ'). Show that H' = {f IX' If E H}. Hint: There is an isometry U: H' ~ H (not necessarily onto) mapping qJ~, to qJx' for each x' E X'. Use U as well as the adjoint operator U*. 3.10. Exercise. If p > 2 then for no A > 0 the kernel exp[ - AIx - yiP] is positive definite on ~ x ~. Nevertheless there are mixtures cp(x, y)

=

LX' exp[ -Alx -

yiP] dj1(A),

Jl E M~(]O, OO[),

which are positive definite.

3.11. Exercise. Let qJ be a positive definite kernel defined on X x X where X is a topological space. Show that qJ is continuous if and only if Re qJ is continuous at each point of the diagonal.

Notes and Remarks Positive (semi-) definite matrices have a long history and we have not traced this history back to its origins. The first instance where a positive definite kernel on a nonfinite set has been considered, seems to be within the theory of integral equations, and the first systematic treatment in this connection was given by Mercer (1909). Mercer defines a continuous and symmetric real-valued function qJ on [a, b] x [a, b] ~ ~2 to be of positive type if and only if

ff

c(x)c(y)cp(x, y) dx dy

~0

(1)

holds for all continuous functions c: [a, b] ~ ~, and he shows that this condition is equivalent to qJ being a positive definite kernel in our terminology; cf. Exercise 1.24. Property (1) had already been singled out a few years earlier by Hilbert (1904) who called the function qJ definite in this case. Mercer also defined a function t/J to be of negative type if and only if - t/J is of positive type. The idea leading to the notion of a negative definite kernel (in our terminology) goes back to Schoenberg (1938b); however, he requires these kernels to vanish on the diagonal-a condition appearing natural in his context: imbedding of metric spaces in a Hilbert space. The product stability of positive definiteness (Theorem 1.12) has been shown by Schur (1911, Satz VII). A remarkable result in this connection was

85

Notes and Remarks

found by Fitzgerald and Horn (1977): if (a jk ) is a positive definite n x n matrix with nonnegative entries then for all real ~ ~ n - 2 the matrix (a lk ) is positive definite, too, and this lower bound for ~ is sharp. The fundamental connection between positive and negative definite kernels expressed in Theorem 2.2 goes back to Schoenberg (1938b) in case the negative definite kernel vanishes on the diagonal. The general case seems to have been proved first in Herz (1962). Infinitely divisible positive definite kernels appear at many places in analysis and-particularly important-in probability theory. Infinitely divisible complex-valued kernels are studied in some detail by Horn (1969a). A study of positive and negative definite kernels with invariance properties under a group action may be found in Parthasarathy and Schmidt (1972). The possibility of representing a positive definite kernel qJ: X x X -+ C as qJ(x, y) = (S) is stable under products and closed in the topology of pointwise convergence. The set &'1 (S) is closed and convex and a base for &'(S), i.e. for every qJ E &'(S)\ {O} there exists a unique pair (A, qJo) with A > 0 and qJo E &'1 (S) such that qJ = ,1qJo, namely A = qJ(e) and qJo = A-lqJ.

1.7. Remark. Let H be a complex Hilbert space and B(H) the set of bounded operators on H. Following Sz.-Nagy (1960) a function : S -. B(H) is called of positive type if for all n E N, all {Sl"'" sn} ~ S and all {~1"'" ~n} ~ H n

L j,k=

0 then (Xf3, max((X, 13) and ak are again absolute values.

90

4. Main Results on Positive and Negative Definite Functions on Semigroups

1.11. Definition. A function J: S -. C is called bounded with respect to an absolute value a (shortly: a-bounded) if there exists a constant C > 0 such that

IJ(s) I ~ Ca(s)

s E S,

for

and J is called exponentially bounded if there exists an absolute value with respect to whichJis bounded. The set of exponentially bounded functions is an algebra.

1.12. Proposition. Let qJ a. Then

E

&J(S) be bounded with respect to an absolute value

IqJ(s) I ~ qJ(e)a(s)

Jor

s E S.

PROOF. Without loss of generality we may assume that I

qJ(S) 12 ~ qJ(S*

0

qJ(e) = 1, so that

s). By iteration we get for n E N I

qJ(s) 12n ~ qJ«s* 0 S)2 n - 1),

and using IqJ(t) I ~ Ca(t) for some constant C > 0 we get 1

qJ(s) 12n ~ Ca«s* 0 s)2 n - 1) ~ Ca(s)2 n ,

hence

IqJ(S) I ~ a(s)

lim C 2 -n

= a(s).

D

1.13. Let qJ E &J(S) and let H 0 be the linear subspace of CS generated by the functions {qJs Is E S}, where qJs(t) = qJ(s* t). Then H 0 is equipped with a scalar product

0

(e)cx(s).

cj cx(x)

By Proposition 1.12 it follows that

"n(s)j'" 2 = g(s* 0 s) ~ g(e)a(s)2

= II f 1 2a(s)2,

which shows that n(s) is a bounded operator on Hoof norm ~ a(s).

D

1.15. Exercise. Let E s denote the shift operator on ([s defined by E s f(t) = f(s t), s, t E S,j' E ([s. Let qJ E &>(S) and let H 0 ~ H ~ ([s be the associated 0

pre-Hilbert space and Hilbert space. Use the closed graph theorem to prove that Es.(H) ~ H if and only if n(s) is a bounded operator on H o. If n(s) is a bounded operator on H 0' then the unique continuous extension n(s) of n(s) to H is given byn(s).f = Es.ffor f E H.

1.16. Exercise. Suppose that S is a group with s* = s- 1. Show that every qJ

E

&>(S) satisfies IqJ(s) I ~ qJ(e), s

E

S.

1.17. Exercise. Let 11(S) denote the set offunctionsf: S -. ([ for which

Ilfli =

L I f(s)1 seS


O. Choose t; > 0 such that t;2qJ(O) < veL). By the Stone-Weierstrass theorem there exists a real-valued function E.9'1 such that ~ 1 on Land 0 ~ ~ t; on supp(Ji). Then 2 E d, 2 ~ 0 and

which is a contradiction. Finally, the set of restrictions to supp(Ji) of the functions in .9'1 is dense in C(supp(Ji)), so by the Riesz representation theorem J.1= v. D The uniqueness statement in Theorem 2.5 combined with Theorem 2.6 can be expressed in the terminology of the theory of integral representations. 2.7. Corollary. Let a be an absolute value on S. Then

£?J'~(S)

is a Bauer simplex.

96

4. Main Results on Positive and Negative Definite Functions on Semigroups

As a special case we consider the absolute value b(s) = 1, s E S. The cone f!jJb(S) is equal to the cone of bounded positive definite functions, and we define

S:= S*

n f!jJb(S) = {p

E

S*llp(s)1 ~ 1 for s E S},

which is a compact subsemigroup of S* called the restricted dual semigroup. It is easy to see that Sis the set of bounded semicharacters. For this special case Theorem 2.6 and Corollary 2.7 lead to the following result :

2.8. Theorem. The set f!jJ~(S) of bounded positive definite functions qJ with qJ(O) = 1 is a Bauer simplex and its set of extreme points is S. A bounded positive definite function qJ has an integral representation cp(s) where J.1

E

=

L

p(s) dl1(p),

SE

S,

M +(S) is uniquely determined.

In the special case of S being an abelian group with s* = - s, Theorem 2.8 reduces to Bochner's theorem for discrete abelian groups, cf. Rudin (1962):

2.9. Theorem. Let G be a discrete abelian group. A function qJ: G -. C is positive definite in the group sense if and only if it is the Fourier transform of a (nonnegative) Radon measure on the compact dual group G. 2.10. The set of signed measures of the form J.11 - J.12 + i(J.13 - J.14) with J.1j E MC+(S*),j = 1, ... ,4 will be denoted MC(S*). Extending the convolution of measures in MC+ (S*) to MC(S*) by bilinearity, MC(S*) becomes an algebra. For /1 E MC(S*) we denote by (t the function

(t(s) =

f

p(s) dJ.1(p),

sES

s*

and {t is called the generalized Laplace transform of J.1. The mapping J.1 H (t of MC(S*) into C S , the generalized Laplace transformation, has the following properties, where a, f3 E C, J.1, v E MC(S*):

(i) (aJ.1

+

f3V)A = a{t

v;

(ii) (J.1 * v) = {t . (iii) {t = 0 => J.1 = 0; A

+

f3v;

i.e. /1 H (t is an injective algebra homomorphism. Since MC+(S*) is mapped onto the cone f!jJe(s), MC(S*) is mapped onto the subspace of C S spanned by f!jJe(s). Among the above properties (i) and (ii) are straightforward to establish, and (iii) follows from the proof in Theorem 2.5. Indeed, if J.1 = (/11 - J.12) + i(J.13 7114)' with J.1j E M C+ (S*), and K s; S* is compact such that supp(J.1 j) S; K, j = 1, ... ,4, then J dJ.1 = 0 for all E d, and the

§2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups

97

restrictions I K form a dense subspace of C(K). By Theorem 2.2.4 it follows that J.1 = o. We identify M +(S) with the set of J.1 E MC+(S*) for which supp(J.1) £; S. For J.1, v E MC+(S*) with support in S, J.1 * v also has its support in S, so we can consider the convolution as a composition in M + (S). In the following result M +(S) is equipped with the weak topology, which is equal to the vague topology. The result is analogous to the Levy continuity theorem in probability theory, but simpler.

2.11. Theorem. The transformation J.1 H P, is a homeomorphism of M +(S) onto q;>b( S). Only the continuity of the inverse mapping p, H J.1 needs some explanation. Let (J.1~) be a net in M + (S), let J.1 E M +(S) and assume p,~(s) -.. P,(s) for each s E S. In particular there exists a o and C > 0 such that J.1~(S) = P,~(o) ~ C for a ~ ao , showing that (J.1~) is eventually in the compact set of measures in M +(S) with total mass ~ C. It therefore suffices to show that J.1 is the only accumulation point for (J.1~). Indeed, if (J is an accumulation point for (J.1~), then p, = fj by the continuity of hence J.1 = (J. D PROOF.

A ,

2.12. Exercise. Let S be an abelian semigroup with involution. Show that the Banach algebra [l(S) from Exercise 1.17 is abelian. Let ~ be the Gelfand spectrum of nonzero multiplicative linear functionals (Rudin 1973, p. 265), and let ~h denote the set of hermitian elements, i.e. the set of L E ~ such that L(f*) = L(f) for f E [l(S). Let r denote the set of nonzero bounded functions y : S --+ C satisfying y(s + t) = y(s) y(t) for s, t E S with the topology of pointwise convergence. For y E r let L y : [l(S) --+ C be defined by Ly(f) =

L f(s)y(s). seS

Show that y H L y is a homeomorphism of r onto ~ which maps S onto ~h. It can be proved that [l(S) is semisimple if and only if r separates the points of S, cf. Hewitt and Zuckermann (1956).

2.13. Exercise. Let (S, +, *) and (T, +, *) be abelian semigroups with involution, and let (S x T, +, *) be the product semigroup defined by (s, t) + (u, v) = (s + u, t + v), (s, t)* = (s*, t*). Show that there exists a topological semigroup isomorphism of S* x T* onto (S x T)*, which maps

S x tonto s;:-r.

2.14. Exercise. Let (S, +, *) and (T, +, *) be abelian semigroups with involutions and let h: S --+ T be a homomorphism. The dual map h: C T -.. C S is defined by h(f) = f h. Show that h(&>(T)) £; &'(S), h(&>e(T)) £; (!Je(s), h(T*) ~ S*, h(T) ~ S and that h* := hi T* is a continuous homomorphism of T* into S* which, furthermore, is one-to-one if h is onto. Let qJ E &>e(s) have the representing measure J.1 E MC+(S*). Show that C h qJ = f h for some f E &>e(T) if and only if J.1 = v * for some v E M +(T*). 0

0

98

4. Main Results on Positive and Negative Definite Functions on Semigroups

2.15. Exercise. Let a be an absolute value on S and let cp E ,qI>~(s) satisfy 2 I cp(s) 1 = cp(s + s*) for all s E S. Show that cp E S*. Show also that if cp E ,qI>~(S) satisfies Icp(s) I = a(s) for all s E S, then cp E S*. 2.16. Exercise. Let (S, +, *) be an abelian semigroup with involution. Show that S* is a linearly independent subset of C S and conclude that if S is finite then card S* ~ card S. Give an example where card S* < card S < 00. 2.17. Exercise. Show that ([ -1, 1], .).. . can be described as T u sgn· T u {sgn 2 } where T := {t 1-+ It I~, a E [0, oo]}, sgn:= 1]0,1] - 1[-1,0[, and Itl o := 1, Itl oo := 1{-1,1}(t) for t E [-1,1]. 2.18. Exercise. Let cp E ,qI>~(S). Then {s E S Icp(s) = a(s)} {s E S II cp(s) I = a(s)} are *-subsemigroups of S.

as well as

2.19. Exercise. Let (S, +, *) be an abelian semigroup with involution and let T:= {p E 511 p I == I}. If cp E ,qI>b(S) has the representing measure J.1 then J.1(T) = infses cp(s + s*). 2.20. Exercise. Let (S, +, *) be an abelian semigroup and consider S u {iiJ} as in Exercise 1.19. Let cp E ,qI>b(S) have the representing measure J.1. Then an extension iP of cp to S u {iiJ} is positive definite if and only if 0 ~ i1J( iiJ) ~ J.1( {Is})· 2.21. Exercise. Let S be an abelian semigroup with involution, let n E N /'.... -. and let S" be identified with (S)" in accordance with Exercise 2.13. Show that if cp E ,qI>b(S") has the representing measure J.1 E M + (5"), and if (J is a permutation of n elements, then cp 0 (J has the representing measure J.1u- I. Conclude that cp E ,qI>b(sn) is symmetric, i.e. cp 0 (J = cp for all permutations (J, if and only if J.1 is symmetric. 2.22. Exercise. Let S = [0, 1] and define Sot = (s + t - 1)+ for s, t E S. Show that (S, 0) is an abelian semigroup with neutral element 1 and absorbing element o. Show further that S is 2-divisible (cf. 6.8) and that S* = {I, 1{1}}.

§3. Negative Definite Functions on Abelian Semigroups In most of this section, we consider negative definite functions on an abelian semigroup with involution (S, +, *). From 3.20 onwards, we will specialize to semigroups with the identical involution.

99

§3. Negative Definite Functions on Abelian Semigroups

For tjJ E %(S) we know by Theorem 3.2.2 that e-tl/J E [!lJ(S) for t > O. Since we only have an integral representation for exponentially bounded positive definite functions and, in particular, for bounded positive definite functions, the following simple result is of some interest.

3.1. Proposition. Let tjJ E %(S). Then e-tl/J E [!lJe(s) (resp. E q;b(S)) for all t > 0 if and only if there exists an absolute value a ~ 1 and a constant C E ~ such that Re tjJ(s)

~

C - log a(s)

(resp. Re tjJ(s)

~

C

If this is the case then C

s E S,

for

s E S).

for

= tjJ(O) can be used.

PROOF. If e-l/J is exponentially bounded (resp. bounded) there exists an absolute value a, where we may assume a ~ 1 (resp. a = 1), such that

Ie-l/J(s) I =

e-Rel/J(s)

~

s E S,

e-l/J(O)a(s),

and the stated inequalities follow. Conversely, if the first inequality of the proposition holds, we get for t > 0

/e-tl/J(S) I ~ e-tCa(sr, which shows that e- tljl absolute value.

q;e(s) (resp.

E

s

E

S,

[!lJb(S) when a

E

=

1) since at is an

D

The set of negative definite functions tjJ for which Re tjJ is bounded below will be denoted %/(S).

3.2. Corollary. For tjJ

E

%/(S) we have Re tjJ(s)

3.3. Proposition. Let tjJ

E

~

tjJ(O),

SE

S.

%/(S) satisfy tjJ(O) ~ O. Then

JltjJ(s+t)I~~+~, PROOF. By Proposition

s,tES.

1.9 the matrix

2 Re tjJ(s) - tjJ(s* ( tjJ(s) + tjJ(t) - tjJ(s

+ s) + t*)

tjJ(s) + tjJ(t) - tjJ(s* + t)) 2 Re tjJ(t) - tjJ(t* + t)

is positive definite, hence, using inequality (1) of §1, I

tjJ(s)

+ tjJ(t)

- tjJ(s*

+ t)

2

1

~ (2 Re tjJ(s) - tjJ(s*

+ s)) + t))

x (2 Re tjJ(t) - tjJ(t* ~

41 tjJ(s) " tjJ(t) I.

Replacing s by s* and extracting the square root gives

ItjJ(s

+ t)1 ~

and the inequality follows.

/tjJ(s)1

+

/tjJ(t)1

+ 2~~, D

100

4. Main Results on Positive and Negative Definite Functions on Semigroups

In some later applications we shall need the following boundedness result for negative definite functions on semigroups with absorbing element (cf. 1.3).

3.4. Proposition. Let S contain the absorbing element w. Then the.functions in JV'(S) are automatically bounded. More precisely if IjJ E JV'(S) and 1jJ(0) ~ 0 then 111jJ1100 ~ fil/J(w). PROOF.

For fixed s

(

E

S the 3 x 3 matrix

1jJ(0)

ljJ(s*)

ljJ(s) ljJ(w)

ljJ(s + s*) ljJ(w)

IjJ(W)) ljJ(w) ljJ(w)

= (ajk)j,k= 1, 2, 3

is negative definite. Writing down explicitly the inequality for C 1 = C 2 = 1, C 3 = -2 gives

L cjC;a jk ~ 0

+ 2 Re ljJ(s) + ljJ(s + s*) ~ 41jJ(w), + s*) = Re ljJ(s + s*) ~ 0 by Corollary 1jJ(0)

l/J(s 3.2 we get 21jJ(w). Similarly, the choice of C 1 = 1, (2 = i, C3 = -1 - i gives - 1m ljJ(s) ~ ljJ(w), and the coefficients C 1 = 1, C 2 = - i, c 3 = i - I yield 1m ljJ(s) ~ ljJ(w). Hence 11m ljJ(s) 1~ ljJ(w) and therefore 1IIjJI 100 ~

and

since

o ~ Re l/J(s)

~

D

fil/J(OJ).

We shall next establish an important relationship between functions JV'(S) and convolution semigroups of Radon measures on the compact semigroup 5, which is the restricted dual semigroup of S.

l/J E

3.5. Definition. Let T be a Hausdorff topological semigroup with neutral element e. A convolution semigroup on T is a family (J.1t)t~O from M~(T) such that t ~ J.1t is a continuous homomorphism from the semigroup (IR+, +) into the semigroup (M~ (T), *), i.e. such that: (i) J.10 = Be; (ii) J.1t * J.1r = J.1t+r for t, r E IR+ ; (iii) t ~ J.1t is weakly continuous. In the case of T = 5 condition (iii) can be replaced by a seemingly weaker one as the following lemma shows.

3.6. Lemma. Let t ~ J.1t be a homomorphism from (IR+, +) into M +(5) such that lim t -+ 0 J.1t

=

B1

weakly, then (J.1t)t ~ 0 is a convolution semigroup on

5.

By Theorem 2.11 it suffices to prove thatjs(t):= fils) is a continuous mapping from IR+ into C for each s E S. By assumption we have js(t + r) = js(t)js(r) and lim t -+ o js(t) = 1, so there exists A > 0 such that Ijs(t) I ~ 2 for t E [0, A]. It follows that Us(t) I ~ 2n for t E [0, nA], in particular js is locally PROOF.

~3.

101

Negative Definite Functions on Abelian Semigroups

bounded. This combined with the equations (t, r, t - r > 0)

Ijs(t

+ r)

- js(t) I = Ijs(t) IIjs(r) -

11, 11

Ijs(t - r) - js(t) I = Ijs(t - r) Iljs(r) -

D

implies that js is continuous.

3.7. Theorem. There is a one-to-one correspondence between convolution semigroups (J.1t)t ~ 0 on S and negative definite functions l/J E %l(S) established

via the.formula (It(S) = e - tl/l(s)

for

t

~

0,

S E

s.

(1)

PROOF. Let (J.1t)t~O be a convolution semigroup. For s E S the mapping

js: t t---+ (It(s) is continuous by Theorem 2.11 and satisfiesjs(t + r) = js(t)js(r), hence of the form js(t) = e-tl/l(S) for a uniquely determined complex number t/J(s). Since e-tl/l E g>b(S) for each t > 0 it follows by Theorem 3.2.2 and Proposition 3.1 that t/J E %(S) and that Re t/J ~ t/J(O). Conversely, if t/J has these properties there exists a uniquely determined family (J.1t)t > 0 from M +(S) such that (1) holds. Clearly

(l,(s)(lr(s) = (It+r(s) for s

E

(J.1t)t~O

S, so

=

lim (It(s)

and

t-O

= 1 = £1(S)

is a convolution semigroup by 2.10, 2.11 and 3.6.

D

We will now consider two special types of functions t/J E %l(S), namely, purely imaginary homomorphisms and nonnegative quadratic forms. 3.8. Definition. A function t/J: S --+ C is called a purely imaginary homomorphism if it has the form t/J = ii, where I: S --+ IR is *-additive, i.e. I(s + t) = I(s) + I(t) and I(s*) = -/(s) for s, t E S. A function q: S --+ IR is called a quadratic form if

+ 2q(t)

2q(s)

=

q(s

+ t) + q(s + t*)

for

s, t

E

S.

If the involution on S is the identical, every purely imaginary homomorphism is identically zero, and a quadratic form is a homomorphism of (S, +) into (IR, +). If S is an abelian group, with the involution being s* = -s for s E S, the functional equation for a quadratic form q is

+ 2q(t)

2q(s)

=

q(s

+ t) + q(s

- t)

for

s, t

E

S.

3.9. Theorem. Purely imaginary homomorphisms and nonnegative quadratic forms belong to %l(S). PROOF. Let

t/J: S

~ Sand {c 1 ,

--+

•.. ,

C be a purely imaginary homomorphism, let {SI' ... , sn} LJ= 1 Cj = O. Then

cn } ~ C with

n

L

j.k=l hence t/J

E

%l(S).

n

CjCkt/J(sj

+ Sk) = L

j,k=1

CjCk(t/J(Sj)

+ t/J(Sk)) =

0,

102

4. Main Results on Positive and Negative Definite Functions on Semigroups

Let q: S --+ [R be a quadratic form. Note that q(O) = 0 and q(s*) = q(s). We define B: S x S --+ [R by

B(s, t) = q(s)

+ q(t)

- q(s

+ t*),

s, t

E

S,

and claim that B has the following properties for s, t, rES:

(i) (ii) (iii) (iv)

B(s, t) = B(t, s); B(s*, t) = B(s, t*) = -B(s, t); B(s + r, t) = B(s, t) + B(r, t); tB(s, s) = lim n oo (q(ns)/n 2 ). -+

Here (i) is clear, and

B(s*, t) = B(s, t*) = q(s)

+ q(t)

- q(s

= q(s + t*) - q(s) - q(t)

=

+ t)

-B(s, t).

To see (iii), we first remark that

+ q(t + t*)

q(s)

=

q(s

+ t + t*)

for

s, t

E

S,

and therefore we have

B(s, t)

+ B(r, t) =

+ q(t) - q(s + t*) + q(r) + q(t) - q(r + t*) = q(s) + q(r) + 2q(t) - t[q(s + t* + r + t*) + q(s + t* + t + r*)] = q(s) + q(r) + 2q(t) - tq(s + r*) - tq(t + t*) - tq(s + r + t* + t*) = tq(s + r) + 2q(t) - tq(t + t*) - t{2q(s + r + t*) + 2q(t) - q(s + r + t + t*)} = q(s + r) + q(t) - q(s + r + t*) = B(s + r, t). q(s)

To see (iv) we remark that for n

E

q(ns) = n 2 q(s) -

N n(n - 1)

2

q(s

+ s*).

This formula is clearly true for n = 1,2, and assuming it is correct for n - 1, n we get

q((n

+

l)s) = 2q(ns)

+ 2q(s)

- q((n - l)s

+ (s + s*))

= (2n 2 + 2)q(s) - n(n - l)q(s + s*) - q((n - l)s) - q(s + s*) = (n + 1)2q(s) -

(n

+

2

l)n

q(s

+ s*).

It follows that

. q(ns) lIm - 2 - = q(s) - tq(s n

-+00

n

+ s*) =

tB(s, s).

103

§3. Negative Definite Functions on Abelian Semigroups

If q is a nonnegative quadratic form it follows by (iv) that B(s, s) s E S. For {Sl' ... , sn} ~ S, {c 1 , ... , cn} ~ 7L we now get n

I

j,k= 1

~

0 for all

n

cjck(q(Sj)

+ q(Sk)

- q(sj

I

+ Sk)) =

j,k= 1

cjckB(Sj' Sk)

B(t, t)

=

~

0,

where

I

t =

CjSj

I

+

j,Cj~O

(-cj)sj.

j,Cjo be the corresponding convolution semigroup on S. For a E S we have r at/J .: t/J E p;>b(S), and if (Ja E M + (S) is the unique Radon measure such that r al/J - l/J = fj a then .

1

(Ja = hm(l - Re p(a)) - Jlt t

t-O

weakly. PROOF. If 1 is the identity operator on CS we have -1 (ra

t

-

1)(1 - e-t"')(s)

= -1 (1 - ra)(e-t"')(s) = -1 t

t

1 s

p(s)(l - Re p(a)) dJlip),

showing that (ra - 1)«1/t)(1 - e- t",)) E p;>b(S). For t -.. 0 this function tends pointwise to rat/J - t/J, which then necessarily belongs to p;>b(S). The assertion about the measure follows immediately from the Levy continuity theorem (2.11). D We now consider an arbitrary hermitian function t/J: S --+ C with the property that r at/J - t/J E p;>b(S) for all a E S, and we denote by (Ja the unique Radon measure on S such that rat/J - t/J = fJa.

3.12. Lemma. Let t/J: S --+ C be a hermitianfunction such that r at/J - t/J E p;>b(S) for each a E S. There exists a unique Radon measure Jl E M + (S\ {I}) such that (1 - R e p( a)) Jl

= (J a I(S\ { 1})

for

a E S.

(2)

104

4. Main Results on Positive and Negative Definite Functions on Semigroups

If tjJ E JVI(S) and (J.1t)t> 0 is the corresponding convolution semigroup then J.1 = lim t.... o(l/t)J.1tl(S\{l}) vaguely. PROOF.

For a E S we let

(!Ja:= {p

E

SIRe p(a) < I},

+

which is an open subset of S. Since (!Ja = {p E Slp(a) I} the family «(!Ja)aeS is an open covering of the locally compact space S\ {I}. On each (!Ja we consider the Radon measure

Since -(ra

-

I)(r b

-

I)tjJ(s) = -(r b

=

L

=

L

I)(r a

-

-

I)tjJ(s)

p(s)(l - Re p(a)) dUb(P)

p(s)(l - Re p(b)) dua(p),

we get by the unicity assertion of Theorem 2.8 that (1 - Re p(a))ab = (1 - Re p(b))aa'

a, b E S,

so the compatibility condition of Theorem 2.1.18 is satisfied for the measures Consequently, there exists a uniquely determined Radon measure J.1 on S\{I} such that J.11 (!Ja = La for a E S, and it is clear that J.1 is the only Radon measure satisfying (2). To finish the proof it is enough by Theorem 2.4.9 to prove that for each

(La)aeSo

aES . 1 1Im-J.1tl(!Ja=L a t .... O

g(p)

:=

vaguely on (!Jao

t

{f(P )(1

- Re p(a)) -

1,

0, is a bounded continuous function on

S, so by 3.11, D

3.13. Definition. Let tjJ be as in Lemma 3.12. The measure J.1 is called the Levy measure for tjJ or for the corresponding convolution semigroup (J.1t)t~O in case tjJ E JV'(S). -

105

§3. Negative Definite Functions on Abelian Semigroups

The Levy measure might have infinite total mass, but (2) implies that

(l - Re p(a)) djl(p)
b(S) for each a E S. (iii) There exists a triple (I, q, J.l) where I: S ~ [R is *-additive, q is a nonnegative quadratic form and J.l E £ such that

ljJ(s) = ljJ(O)

+ il(s) + q(s) + I. JS\{l}

for all s E S.

(1 - p(s)

+ iL(s, p»

djl.(p)

109

§3. Negative Definite Functions on Abelian Semigroups

The triple (I, q, J.l) is uniquely determined by t/J measure for t/J and _ l' Re t/J(ns) 1m 2

q(s ) -

n

" _ 00

E

%'(S), J.l being the Levy

+ I'1m t/J(n(s + s*» " -+

2n

00

,

SE

S.

PROOF, "(ii) => (iii)" Let J.l be the Levy measure for t/J and let t/J Jl be defined as in Proposition 3.18. The function

r := t/J - t/J(O) - t/J Jl is hermitian and rar - r = fi a - (O"al(S\{I}»A = O"a({1}) for a E S, hence positive definite and constant. By Lemma 3.14 it follows that t/J has the representation stated in (iii). Finally, if (iii) holds, it is clear by 3.9, 3.14 and 3.18 that t/J E %'(S), and that the Levy measure for t/J is J.l. This shows that the triple (I, q, J.l) is uniquely determined by t/J, and we find

= t/J(?) + q(~s) +

Re t/J2(ns)

n t/J(n(s

n

+ s*» =

n

t/J(O) n

n

~ (1

f

_ Re(p(s)"» dJ1.(p),

JS\{l} n

+ q(n(s + s*» + f

JS

n

\{l}

! (l

_ Ip(sW") dJ1.(p).

n

By the proof of Theorem 3.9 we have

,

hm "-+00

q(ns) -2-

n

= q(s) - tq(s + s*),

and by the functional equation for quadratic forms

q(n(s

+ s*» = nq(s + s*),

which gives the formula for q(s) provided the integrals tend to zero. That this is true follows from the dominated convergence theorem because of the inequalities

1 n2 (1 - Re(p(s)"»

~

n2

4

(1 - Re p(s»,

1 -(1 -lp(s)1 2 ") ~ 1 - Re p(s

n

Putting p(s)

= re i8 , r E [0, 1], 0 E ]

1

n2 (1 - r" cos(nO» Clearly 1 - r"

~

~

n2

4

-

+ s*).

n, n], these inequalities become n EN.

(1 - r cos 0),

n(1 - r) for r E [0, 1], n E N, and using the inequalities

ISi: x I ~ 1,

X E ~,

sin x 2 --2x

- n

for

n

Ixl -~ -2'

110

4. Main Results on Positive and Negative Definite Functions on Semigroups

we find for f)

E ]

-n, n]

.

= 2 SIn

1 - cos(nf)

2

nf)

2 =

hence

1

2 (1 -

n

r" cos(nf))

1

~ 2

n

(1 - r")

1

~ ;; (1 -

r)

~

+2

n

(1 - cos nO)

n2

+ 4 r"(1 -

cos f) ~

n2

4

(1 - r cos f).

D

Remark. In the above representation q and J1 are independent of the Levy function L, whereas I depends on the choice of L. In the rest of this section, we assume that S is an abelian semigroup with the identical involution. Then negative definite functions are real-valued and %'(S) is the set of negative definite functions t/J which are bounded below or equivalently satisfy t/J(s) ~ t/J(O). For f E [Rs and a E S, we define ~af E [Rs by ~af(s)

= f(s + a) - f(s),

s E S,

hence ~af = (ra - I)fwith the previous notation. The function L == 0 is a Levy function for S, and from the previous results it is easy to get the following: 3.20. Theorem. Let S have the identical involution. The following conditions are equivalent for a function t/J: S -+ [R: (i) t/J E %'(S). (ii) ~a t/J E pjJb(S) for each a E S. (iii) There exist an additive function q: S J1 E M + (8\ {I}) such that

t/J(s)

= t/J(O) + q(s) +

-+

r

JS

[0, oo[ and a Radon measure

(l - p(s)) dJ1.(p).

\{l}

The decomposition in (iii) is uniquely determined, J1 being the Levy measure for t/J and q (s)

.

= 11m "-+00

t/J(ns) --, n

SE

S.

If q == 0 then t/J is bounded if and only if J1(S\ {I})
O. Furthermore, (.pJlyn)(z)

= {OO (-a)"e- za dJl(a)

for

Re z > 0,

n

~

0,

so in particular for

x >

o.

We have the following corollary of Proposition 4.2.

4.5. Corollary. A function qJ: IR + -+ IR is continuous, positive definite and bounded &(' and only if it is the Laplace transform ofa measure in M~(IR+). In Theorem 6.13 we will give a completely different characterization of the functions in Corollary 4.5.

4.6. S = (IR~, +). For s = (Sl' .•• , Sk) E IR~ and a = (a 1 , .•• , ak ) E [0, oo]k we put <s, a) = L~=l sja j and define Pa: IR\ -+ IR by Pa(s) = e-(s,a). Then

115

§4. Examples of Positive and Negative Definite Functions

the mapping a ~ Pa is a homeomorphism and an isomorphism of ([0, ooJ k, +) onto (8, .). The semicharacter Pa is continuous on [R~ when a E [R~ and discontinuous when a E [0, 00 Jk\[R~. Clearly we have results analogous to 4.2 and 4.3. For J.l E M~([R~) the function S E [R~

is the (k-dimensional) Laplace transform of J.l. This formula makes sense for s E e~ and defines a continuous function which is holomorphic in {z E eiRe z > O}k. A k-dimensional version of Corollary 4.5 is

4.7. Proposition. A function 0, we get the following partial solution of the complex moment problem from Theorem 2.6.

N5

4.12. Proposition. A complex-valued function cp is positive definite on (~~, +, *) and verifies

Iq>(n, m)1 ~ Ca n + m

for

(n, m) E ~5,

where C, a > 0, if and only if q> is a complex moment function of a measure E M +({z E Clizi ~ a}), i.e. is of the form

Il

({)(n, m)

=

r

J1z1 ~a

z"zm dj1(z),

(n, m)

In particular q> is positive definite and bounded moment function of a measure j.1 E M + (D).

E

~5.

if and only if cp is a complex

In preparation for the Levy-Khinchin representation in this case we need the following two lemmas. 4.13. Lemma. The *-homomorphisms I: (~5,

l(n, m) = a(n - m), and the quadratic forms q: (~5, +, *) q(n, m) = a(n

+ m) +

0 ~land only ~la, fJ ~

-+

+, *)

-+

a

IR,

E

IR are given by

IR are given by

fJ(n - m)2,

a, fJ

E

IR.

o.

Here q

~

PROOF.

The assertion about the *-homomorphisms is easily established. Let

q be a quadratic form and put a = q(1, 0), b = q(1, 1). Since q(s q(s) + q(t) if t = t* we have q(n, m)

=

q(n - m,O)

+ mb

if

n ~ m,

+ t) =

118

4. Main Results on Positive and Negative Definite Functions on Semigroups

and from the proof of Theorem 3.9 we know that 2

q(n, 0) = n a For n

~

n(n - 1) 2 b,

n

~

o.

m we therefore have q(n, m)

=

(n - m)2

(a - ~) + (n + m) ~'

and since q(n, m) = q(m, n) this holds for all (n, m) E N5· Conversely, it is clear that any function of the form q(n, m) = a(n + m) + f3(n - m)2, a, f3 E lR, is a quadratic form, which is nonnegative if a, f3 ~ 0; and if q ~ 0 then q(l, 1) = 2a ~ 0 and lim n ---. oo (ljn 2 )q(n, 0) = f3 ~ o. D

4.14. Lemma. Thefunction L:

~5 x

D -+ lR given by

L«n, m), z) = y(n - m),

Z

= x + iy

+, *).

E

M + (D\ {I}) satisfies (3) of

is a Levy function for (~5, 3.13 if and only if

r

A measure J.1

(1 - x) dJ1(z)
0 and Z = x + iy E D we have

1 - Re(z"zm)

= 1-

L (;)(;)x"+m- p- q( -1)q( -1)(P+Q)/2 y P+q,

where the sum is over (p, q) such that p ~ n, q ~ m and p + q is even. For p = q = 0 we have the term x n + m • In all other terms we have a factor yp+q, where p + q is even ~ 2, so yp+q ~ (1 - x 2 )(p+q)/2. Therefore

o~

1 - Re(z"zm)

~

1 - x"+m

+L

(;)(;)(1 -

x 2)(p+q)/2,

which shows that there exists a constant K > 0 (depending on n, m) such that 1 - Re(znzm) ~ K(l - x) so (3) of 3.13 holds. (This result also follows from Exercise 3.28.) The function L clearly satisfies (i) and (ii) of 3.17, and (iii) follows if we establish

11 -

znzm

+

i(n - m)yl

~

K(1 - x)

for

z

= x + iy E D

and a suitable constant K depending on (n, m). This is easily seen for n + m and for n + m ~ 2 we find 1-

znzm

+

i(n - m)y = 1 - x n + m

+

+ y2p(X, y),

iy(n - m)(1 - x n + m -

1

)

~

1

119

§4. Examples of Positive and Negative Definite Functions

where P is a polynomial, hence 11 -

znzm

+ i(n

~

- m)yl

(1 - x n + m )

+ In -

+ (1

max IP(x, y)1

- x

2

)

ml(1 - x n + m -

1

)

D

D

and the inequality follows. We can now give the Levy-Khinchin representation. 4.15. Proposition. A function has a representation

t/!(n, m) = t/!(O, 0)

t/!: N6 -+ C

+ ia(n

+ f

- m)

(1 - z"zm

belongs to JVI(N6)

+ b(n + m) + c(n

+ iy(n

-

m»

if and only if it

- m)2

dj1(z),

JD\{l}

where a E [R, b, c

~

0 and J1

E

M + (D\ {I}) satisfies

f

(1 - x) dJ1(z)
(S) is nonnegative, decreasing and bounded, so, in particular, &>(S) = &>b(S). In fact qJ(s) = qJ(s + s) ~ 0, and if S ~ t then qJ(t)2 = qJ(s + t)2 ~ qJ(s + s)qJ(t + t) = qJ(s)qJ(t) so qJ(t) ~ cp(s). In

120

4. Main Results on Positive and Negative Definite Functions on Semigroups

particular, qJ(t) ~ qJ(O). Similarly, ljJ ljJ(O) ~ ljJ(s) so %(S) = %1(S).

E

%(S) is increasing and satisfies

By specialization of the Theorems 2.8 and 3.20 we get:

4.17. Proposition. Let (S, +) be an idempotent semigroup. For qJ E &>(S) there is a unique j.1 E M + (Y') such that

qJ( s) andfor ljJ

E

=

j.1( {I

E

Y' IS E I}),

%(S) there is a unique j.1

ljJ(s)

=

ljJ(O)

SE

EM +(Y'\{S})

+ j.1({I E Y'\ {S} Is $ I}),

S,

such that SE

S.

A function qJ: S ---+ [0, oo[ which is decreasing and bounded need not be positive definite, cf. Exercise 4.25. However, if S has the further property of the order being total, i.e. for any s, t E Seither S ~ t or t ~ s, then we have

4.18. Proposition. Let (S, +) be an idempotent semigroup for which the ordering is total. Then qJ: S ---+ [R is positive (resp. negative) definite if and only if qJ is nonnegative and decreasing (resp. increasing). PROOF. Let qJ: S ---+ [0, 00 [ be decreasing and let s l' , Sn E S, C l' ... , Cn E ~. By 3.1.2 it is no restriction to assume s 1 ~ S2 ~ ~ Sn. From probability theory we know the existence of n independent normally distributed random variables X l' . . . , X n with mean zero and variances qJ(SI)' qJ(S2) - qJ(SI)' . .. , qJ(sn) - cp(sn - 1). (If one of the variances is zero the corresponding normal distribution is degenerate.) Put ~ = L~= 1 Xj' k = 1, ... , n and denote the expectation by IE; then ifj ~ k,

IE( lj ~)

= IE( YJ) = qJ(S 1) + qJ(S2) - qJ(S 1) + ... + qJ(Sj) - qJ(Sj_ 1) = qJ(Sj) = qJ(Sj + Sk)

implying

If qJ: S ---+ [R is increasing then e - tep is nonnegative and decreasing for each t > 0, and it follows by Theorem 3.2.2 that qJ is negative definite. D

4.19. As an example of an idempotent semigroup whose ordering is total we consider S = [0, 1], the composition being maximum. The neutral element is zero and 1 is an absorbing element. The ordering of 4.16 is the usual ordering. The set Y' of Proposition 4.17 has the following elements: {O}, [0, a], [0, a[ where a E ]0, 1], and we will now describe the topology on Y' induced by the isomorphism between S* and !/.

121

§4. Examples of Positive and Negative Definite Functions

a'" [0, a]

b '" [0, b [ Figure 1

A base of neighbourhoods of {O}, [0, a] and [0, bE, where a, bE ]0, 1], is given, respectively, by

°

(i) {{o}} u {CO, s] Is E ]0, e[} u {CO, s[ Is E ]0, 8[} for < 8 < 1; (ii) {CO, S]ISE [a, a + 8[n[0, I]} u {CO, s[lsE]a, a + 8[n[0, I]} for < 8 < 1; (iii) {CO, s] Is E ]b - 8, b[n[O, I]} u {CO, s[1 s E ]b - 8, b]n]O, I]} for < 8 < 1.

° °

We remark that f/ is totally ordered under inclusion and that the corresponding order topology on f/ is the topology just described. It is useful to think of f/ as two copies of [0, 1] glued together in the point as Figure 1 illustrates. On the upper segment a E ]0, 1] represents [0, a] and on the lower segment b E ]0, 1] represents [0, be. We know that f/ is a compact Hausdorff space and see that [0, 1] is an isolated point. Furthermore f/ is one of those strange nonmetrizable spaces which are first countable (i.e. every point has a countable neighbourhood base). In fact, if f/ was metrizable, then a classical theorem from topology (cf. Bauer (1978, p. 217)) implies that the Banach space C(f/) of continuous functions on f/ is separable, which is not true: For a E [0, 1] let ba E C(f/) be the function I ~ l I (a). Then Ilb a - bbll = 1 for a =+= b, so the open balls B(b a , t) in C(f/), a E [0, 1] form a noncountable family of disjoint open sets contradicting the separability. As an application of 4.17 and 4.18 we get

°

4.20. Proposition. To every decreasing function qJ: [0, 1] exists a unique Radon measure J1 on f/ such that qJ( s)

= J1( {I

E

f/ Is

E

I}),

S E

-+

[0, oo[ there

[0, 1].

4.21. Exercise. Let (N, .) be the multiplicative semigroup of natural numbers. Show that the dual semigroup and the restricted dual semigroup are isomorphic to IR lPl and [-1, l]lPl where lP = {2, 3, 5, ...} denotes the set of primes.

122

4. Main Results on Positive and Negative Definite Functions on Semigroups

4.22. Exercise. Let qJ E g>b(N o) have the representing measure J.1 on [ - 1, IJ. Show that L~ I qJ(n) 1 < 00 if and only if(1 - Itl)-l E 2 1(J.1). 4.23. Exercise. Let A = (au) be a real positive definite k x k matrix. Show that there exists a function qJ E g>b(Nt) such that for

i,j=I, ... ,k,

where e 1 = (1,0, ... ,0), ... , ek = (0, ... ,0,1). Hint: Assume laul ~ 1. There exists a real k x k matrix B = (b ij ) such that A = B*B. Let Xl' ... ' X k E IRk be the row-vectors of B, and define a measure /l E M +([ - 1, IJk) by putting mass 1 at each of the points X l' . . . , X k .

4.24. Exercise. Let qJ: [0, IJ -+ [0, 00 [ be decreasing and let J.1 be the representing measure on f/ according to Proposition 4.20. Show that qJ(a) - qJ(a

+ 0) =

°

~ a < 1,

/l( {CO, aJ}),

O 0, B is the at most countable set of b E JO, IJ such that p( {CO, b[}) > and J.1c is the continuous part of p. The function s ........ /lc( {I E f/ IS E I}) is continuous and decreasing.

°

4.25. Exercise. Let X be a nonempty set. The set IP(X) of subsets of X is an idempotent semigroup under intersection. Describe the semicharacters of IP(X) explicitly in the case where X has two elements, and show that qJ = 1 - 1{0} is decreasing but not positive definite. 4.26. Exercise. On IR + we define X 0 Y = X + Y + xy. Show that (IR +, 0) is an abelian semigroup and that x ........ log(1 + x) is an isomorphism of (IR+ , 0) onto (IR+, +). Show that the bounded semicharacters on (IR+ ,0) are given by x ........

(_1 )t + 1

x

for

°

~ t ~

00.

4.27. Exercise. An infinite matrix (a jk ) of the form ajk = qJ(j + k) with qJ: No -+ IR is called a Hankel matrix. It is positive (resp. negative) definite if and only if qJ is positive (resp. negative) definite on the semigroup (No, +), and it is infinitely divisible if and only if qJt is positive definite for all t > 0. Show that the so-called Hilbert matrix (1/(j + k + 1))j,k~O is infinitely divisible. -

123

§5. r-Positive Functions

4.28. Exercise. Let t/J: IR + -+ IR + be given. Show that if qJ 0 t/J E g>b(1R +) for all qJ E g>b(1R +).

4.30. Exercise. For M equivalent:

£;

t/J E LN(IR +) if and only

M~ (IR~) the following three conditions are

(i) M is relatively compact with respect to the weak topology. (ii) {fe III Il E M} is uniformly equicontinuous on IR~ . (iii) {fellill E M} is equicontinuous in O.

Hint: Use Prohorov's compactness criterion mentioned in the Notes and Remarks to Chapter 2. 4.31. Exercise. (Continuity Theorem for Laplace Transforms). Let a sequence Ill' Ill'··· E M~(IR~) be given such that qJ(t) = lim n_ oo (feJ1n)(t) exists for all t E IR~ and assume that qJ is continuous at O. Then qJ = feJ1 for some Il E M ~ (lR k+) and Iln -+ Il weakly. 4.32. Exercise. If M ~ (IR~) is equipped with the weak topology and g>~ (IR~) n C(IR~) with the topology of uniform convergence on compact subsets of IR~, the Laplace transformation fe: M ~ (IR~) -+ g>~ (IR~) n C(IR~) is a homeomorphism.

§5. r-Positive Functions In this section, we shall present an approach due to Maserick (1977) to the integral representation of positive definite functions. Let A be a real or complex commutative algebra with identity e and involution *. If A is a real algebra we always assume a* = a for all a E A. The basic idea is to find an integral representation of certain "positive" linear functionals on A, the representing measure being concentrated on a set of multiplicative linear functionals. The connection to semigroups is obtained if A is chosen to be the algebra of shift operators generated by the shifts E s' s E S, where (Esf)(t) = f(s + t) forf E C S . By the relation L(Es) = f(s) functionsfon S are in one-to-one correspondence with linear functionals Lon A. The positivity concept for linear functionals depends on the notion of an admissible subset r of A. 5.1. Definition. A nonempty subset r

(i) a* = a for all a E r; (ii) e - a E alg span+(r) for all a E r; (iii) alg span(r) = A.

£;

A is called admissible if

124

4, Main Results on Positive and Negative Definite Functions on Semigroups

Here alg span(r) (resp. alg span +(r)) is the set of linear combinations ~ 0) and each Xi is a finite product of elements from r. Note that alg span +(r) is the smallest convex cone in A, stable under multiplication and containing r. Furthermore e E alg span+(r) because of (ii).

I?= 1 AiXi, where each Ai is a scalar (resp.

5.2. Lemma. Let r be an admissible subset of A. Then: (i) e - Xl ... Xn E alg span +(r)for Xl' ... ' Xn E r; (ii) for X E alg span +(r) there exists 8(X) > 0 such that e for 8 E [0,8(X)]. PROOF.

8X E

alg span +(r)

The assertion (i) follows from the algebraic identity

"n

n

n

n

e-

=

Xj

(1'( XjJ

~

e-

(1)

X j )1-(1'J,

(1$1 j=l

j=l

where the summation is taken over all a = (a l ' ... , an) E {O, 1}n\ {(I, ... , I)} (note that XO := e), and this identity follows in turn by the formula n

e=

n(x

j

+ (e

-

Xj)).

j= 1

If X = LJ= 1 Ai Yj' where A j > 0 and Yj is a product of elements from r, we put 8(X) = (IJ= 1 Aj ) - 1. For 8 E [0, 8(X)], we find

e-

8X

=

(1 -

(8 8 X

))e + e .± Aie- Yj)

E

alg span+(r)

J= 1

D

because of (i). A linear functional L: A

L(a)

~ 0

-+

C is called r-positive, where r is admissible, if

for all

a E alg span +(r).

This holds if and only if

L(a 1

•••

an)

~

0

{a 1 ,

for all finite sets

••• ,

an}

~

r.

The set Ai of r-positive linear functionals on A is a convex cone in the algebraic dual space A*, closed in the topology a(A*, A). Note that L(x*) = L(x) for X E A when L E Ai. By Lemma 5.2 it follows that L(e) > 0 for L E Ai\ {O}, so

B:= {L is a base for

E

AiIL(e) = I}

Ai.

5.3. Lemma. The base B is compact. PROOF.

For

X E

A there exists a constant K x > 0 such that

for

L

E

Ai.

~5.

125

r-Positive Functions

In fact, if x E alg span +(r) we can use K x = 8(X)-l with 8(X) from 5.2(ii), and everyxEAcanbewrittenx = Xl - X 2 + i(x 3 - x 4 )withx j Ealgspan+(r), j = 1, ... ,4. If (L(X) is a universal net on B the above inequality shows that L(x) := lim(X L(X(x) exists for every x E A, and this implies the compactness

D

ofB.

Let ~ denote the set of r-positive, multiplicative linear functionals on A, which are not identically zero. Clearly ~ is a compact subset of B.

5.4. Theorem. A linear functional L on A is r-positive ~{and only ({there exists a (necessarily unique) Radon measure J.1 E M +(~) such that L(x} =

i

l5(x} dJ1(l5}

for

The base Bfar Ai is a Bauer simplex with ex(B) =

x

A.

E

(2)

~.

A functional of the form (2) is clearly r-positive. Given L E Ai, the existence of J1 such that (2) holds follows from Theorem 2.5.6, once we have shown that ex(B) £; ~. Suppose L E ex(B) and define for a E A the translate La of L by La(x) = L(ax). For a E alg span+(r) we have La E Ai. Since L = La + L e - a and La' L e - a E Ai for a E r, La is proportional to L. The set of elements a E A for which La is proportional to L is a subalgebra of A containing r, hence La = AaL for all a E A. Evaluating at x = e shows Aa = L(a), hence L E ~. Defining y: A ---+ A by y(x) = xx* an application of Corollary 2.5.12 yields ex(B) = ~. The set of functions £5 ~ £5(x), x E A, is a point-separating *-subalgebra of C(~, C) containing the constant functions, hence uniformly dense in C(Ll, C) by the Stone-Weierstrass theorem. This implies that the representing measure is uniquely determined. D PROOF.

From the integral representation in Theorem 5.4 we immediately get:

5.5. Corollary. Any functional L for all x E A.

E

Ai is positive in the sense that L(xx*)

~

0

It is interesting to note that the corollary can be proved directly without use of the integral representation. This was done by Maserick and Szafraniec (1984) as follows: For a function .1': [0, 1] ---+ C and x E A we define

Bn(x;f) = kt/(~)(;)xk(e - x)"-k, and for f'(t) = 1, t, t 2 we find (cf. Davis (1963, p. 109))

Bn(x; 1) = e,

Bn(x; t) = x,

1 Bn(x; t 2 ) = - X n

n- 1

+ - - x2• n

126

4. Main Results on Positive and Negative Definite Functions on Semigroups

Similarly, for a functionf: [0,1]2

-+

C and Xl' X2 E A we define

Bn(x x ;f) = j.~/(~' ~)(;)(:)x{(e - x1)n- jx~(e - x )n-k, 1,

and for f(s, t)

2

=

2

(s - t)2, we find

Bn(x l' x 2 ; (s - t)2) = Bn(x l' X 2 ; S2)

+ Bn(x l' X 2 ; -

= Bn(x l ; S2) - 2B n(x l ; s)B n(x 2 ; t)

= (Xl -

If Xl' X2 E A such that Xl' LE Ai

X 2)

X2'

2

+ -1 (Xl + X2 n

-

+ Bn(x l' X 2 ; t 2) + Bn(x 2 ; t 2)

2st)

2

Xl -

2

X 2 )·

e - Xl' e - X2 E alg span+(r), we have for

hence for n -+ 00: L«x l - X 2)2) ~ O. For Xl' X2 E alg span +(r) there exists 8 > 0 such that 8X l , 8X 2 , e - 8X l , e - 8X 2 E alg span +(r), cf. 5.2, and therefore L(8 2 (X l

-

X2)2) ~ 0,

so L«x l - X2)2) ~ O. An element X E A can be written X = Xl - x 2 + i(X3-X4) with xjEalgspan+(r), j=1, ... ,4, so xx*=(X l -X 2)2+ (x 3 - X 4 )2 and finally L(xx*) ~ o. Let S be a commutative semigroup with involution, and let

A

=

{.f

cjEsjln EN, cj

)=1

E

C, S} Sj

E

be the algebra of shift operators on C S . We recall that Esf(t) = f(s + t) for s, t E S, f E ([s. Clearly, A is a commutative algebra with unit I = Eo, and defining

we get an involution on A. Since (Es)ses is a basis for A, functionsf: S -+ C and linear functionals L: A -+ C are in one-to-one correspondence via the formula L(E s ) = f(s). Iffand L correspond to each other and n

L cjEsj E A

T =

j= 1

then n

L(T) =

L cjf(Sj) = j= 1

Tf(O).

127

§5. r-Positive Functions

It follows that n

L cjCkf(sj + st),

L(TT*) =

j,k= 1

so L is positive if and only iff E (!P(S). If r ~ A is admissible, we call f r-positive if the corresponding linear functional L is r-positive, i.e. if and only if

r.f(O)

~

for all

0

T

E

alg span + (r),

or if and only if

T1 ... T"f(O)

~

0

for all

T1 ,

•• • ,

T" E r.

Note that a semicharacter p is r-positive if and only if Tp(O) T E r, since T1 ... T"p(O) = T1P(O) ... T"p(O).

~

0 for all

5.6. Theorem. Let r be an admissible subset ofthe algebra A ofshift operators.

Every r-positive function f: S -+ C is positive definite, exponentially bounded and has an integral representation f(s)

=

f.

p(s) dJ1(p),

S·

where J1 E M +(S*) is concentrated on the compact set of r-positive semicharacters. PROOF. By Theorem 5.4 the linear functional L corresponding to the rpositive functionfhas a representation

L(T)

=

i

(j(T) dj1«(j),

TEA,

where fi E M +(~). For b E ~ the function s H b(E s ) is a r-positive semicharacter, and the mapping j: ~ -+ s* given by j(b)(s) = b(E s ) is a homeomorphism of ~ onto the compact set j(~) of r-positive semicharacters. The image measure J1 := fij of fi under j is a Radon measure on S* with compact support contained in j(~), and replacing T by Es we get

f(s) =

f.

p(s) dJ1(p),

s

E

S.

S·

This shows thatfis exponentially bounded and positive definite.

D

5.7. A subset G ~ S is called a generator set for S, if every element in S\ {O} is a finite sum of elements from G u {a* Ia E G}. Let ~: S -+ ~+ be an absolute value such that ~(a) > 0 for all a E S. For (J E C, a E S, we define

l(

a) n... a = "2 I + 21X(a) Ea + 21X(a*) Ea· · (J

128

4. Main Results on Positive and Negative Definite Functions on Semigroups

The family

=

rG

{Qa,alaE

{±1, ±i},aEG}

is easily seen to be admissible if G is a generator set. The following converse of Theorem 5.6 holds:

Iff is positive definite and exponentially bounded there exists an admissible r such that f is r-positive. In fact, there exists an absolute value tJ, such that f is tJ,-bounded, and we may assume tJ,(a) > 0 for all a E S. Then

f(s) =

f

pes) djJ.(p),

s E S,

where J.1 E M + (S*) is supported by the compact set of tJ,-bounded semicharacterse With r = rG as above, where G is a generator set, we get

na,af(s) =

f

p(s)t[l

+ lX(a)-l

Re(O"p(a))] djJ.(p),

and since Ipea) I ~ tJ,(a) for p E supp(J.1) it follows that Qa, a f E (!lJCX(S). Hence (!lJCX(S) for a j E {± 1, ±i}, aj E G, in particular,

Qa1,a1 ... Qan,anf E

Qa1,a1 ... Qan,anf(O) ~

showing thatfis rG-positive. Using the absolute value tJ,

==

1 we get" (i)

=>

0 (ii)" of the following result:

5.8. Proposition. F or a function f: S -+- C the following conditions are equivalent: (i) f E (!lJb(S); (ii) f is r-positive, where r

= {Q a , a I a E {± 1, ± i}, a E S} and Qa, a

=

1(I + 2a E + 2(j) E

2"

a

a", .

PROOF. To see" (ii) => (i)" it suffices by Theorem 5.6 to verify that a r-positive semicharacter is bounded. But p E S* is r-positive if and only if 1 + Re(ap(s)) ~ 0 for s E S, a E {± 1, ± i}, i.e. if and only if Re pes), 1m pes) E [ -1, 1] for all s E S, which is equivalent with pES. D

5.9. Remark. In the above proposition it is not possible to replace r by rG

=

{Qa,ala E

{± 1, ±i}, a E G},

where G is a generator set for S. The implication" (i) => (ii)" is, of course, still true, but" (ii) => (i)" might fail to hold. In fact, if S = (N6, +, *) is the semigroup studied in 4.11, then G = {(I, O)} is a generator set and the

129

§6. Completely Monotone and Alternating Functions

rG-positive functions are given by f(n, m)

=

r

znzm dJl(z),

(n, m)

E

)[-1,1]2

N~,

which is a larger class than q;b(S). In the case where the involution is the identical this remark does not apply:

5.10. Proposition. Assume S has the identical involution and let G be a generator set for S. Then the following conditions are equivalent: (i) f E q;b(S); (ii) (1 ± Eat) ... (1

± Ean)f(O)

~ 0

for all aI' ... , an E G, n E N.

It is clear that" (i) => (ii)" holds, and (ii) is equivalent with f being r-positive, where

PROOF.

r = {tel

± Ea)/a E G}

is admissible. Finally every r-positive semicharacter p is bounded, because Ip(a)1 ~ 1 for a E G implies Ip(s) I ~ 1 for all s E S. Note that Q± 1, a = ± Ea) and Q± i, a = tl, and the families

tel

{t(l±Ea)laEG}

and

{Qa,alaE{±l,±i},aEG}

D

lead to the same r-positive functions.

5.11. Exercise. Let S = (N~, +) and let A(k) be the algebra of real polynomials in k variables Xl' •.• , X k . Show that the set r =

{Xl' ... ,

xk , 1 -

Xl -

... -

xk}

is admissible. There is a one-to-one correspondence between functions

f: N~ ~ ~ and linear functionals L: A(k) ~ ~ established via L(x n ) = fen), n E N~. Show that f is r-positive (in the sense that the corresponding linear functional L is r-positive) if and only if there exists J.1 f(n)

=

L

n

x dJl(x),

E

M +(K) such that

n E N~,

§6. Completely Monotone and Alternating Functions In this section, S is an abelian semigroup with the identical involution. We have already introduced the shift operator Ea: ~s -+ ~s defined by Eaf(s) = f(s + a) and ~a = Ea - 1, where 1 denoted the identity operator. Since {E a Ia E S} is a commuting family of operators, the algebra generated by this

130

4. Main Results on Positive and Negative Definite Functions on Semigroups

family is commutative, too; in particular, we have for

a, b E S.

Sometimes it is convenient to use also the operators

Va:=I-Ea=-L\a'

aES,

which are, of course, again commuting; note that Va Vb

n [1 -

= L\aL\b' and that

n

Val···

VanP(S) = p(S)

(1)

p(a i )]

i= t

for p E S*. In his fundamental work on capacities Choquet (1954) studied functions on S which he called monotone (resp. alternating) 0.[ infinite order. Here we shall call these functions completely monotone (resp. completely alternating).

6.1. Definition. A function qJ: S -+- ~ is called completely monotone if it is nonnegative and if for all finite sets {at, ... , an} ~ Sand S E S Val ... Van qJ(S) ~ O. A function and S E S

t/J: S

-+- ~

is called completely alternating if for all {at, ... , an}

~

S

The set of completely monotone (resp. alternating) functions is denoted .A(S) (resp. d(S)). It is clear that .A(S) and sl(S) are closed convex cones in ~s, and the nonnegative (resp. real) constant functions are contained in .A(S) (resp. scl(S)). If qJ E .A(S) then EaqJ E .A(S) and, similarly, if t/J E scl(S) then Eat/J E d(S) for all a E S, since Val ... Van(Eaf) = E aV a1 ... Vanfforf E ~s and since E a is a positive operator. Notice that qJ E Jt(S) and t/J E d(S) satisfy 0 ~ qJ ~ qJ(O) and t/J ~ t/J(O).

6.2. Remark. The following terminology is frequently used for s, at, ... , an E S: Vtf(s; at) = f(s) - f(s and inductively for n

~

+ at),

2

Vn.[(s; at, ... , an) = V n-1.[(s; at, ... , an-t) - Vn-l.[(s

+ an; at,···, an-t).

The following result connects the two concepts of completely monotone and alternating functions, and it is easily established.

131

§6. Completely Monotone and Alternating Functions

6.3. Lemma. A function t/!: S -+- ~ belongs to deS) a E S the function ~a t/! belongs to vII(S). PROOF.

1ft/!

E

deS) then L\at/! = -Vat/!

if and only if for every

~ 0 and

Val··· Van(L\at/!) = ~a Val··· Vant/! = - Va Val··· Vant/! ~ O. On the other hand, if ~a t/!

E

vII(S) for all a E S, then

Val··· Vant/! = -Val··· Van_I(L\ant/!) ~ O.

D

The relation of completely monotone functions to r-positive functions is described in the following:

6.4. Theorem. Let G ~ S be a generator set. The family r = {E a , 1 - Ea Ia E G} is admissible, and for a function qJ: S -+- ~ the following conditions are equivalent: (i) qJ is completely monotone. (ii) qJ is r-positive. (iii) There exists a measure J.1 E M +(S +) such that cp{s)

=

f.

pes) dJl{p),

Js

s

E

S.

+

(I - Ea) ... (1 - Ean)E SI ... Esrn qJ(O) = Val ... Van qJ(S 1

+ ... + Sm)

so (i) => (ii). The implication "(ii) => (iii)" follows from Theorem 5.6 since P E s* is r-positive ifand only if 0 ~ pea) ~ 1 for a E G, but this is equivalent with pES + because every s E S\ {O} is a finite sum of elements from G. Finally, if (iii) holds, then qJ(s) ~ 0 and by (1) Val'"

so qJ

qJ

E

vII(S).

Vancp{s)

=

f.

Js

p{s).n (1 - pea)) dJl{p) +

J= 1

~ 0, D

A function qJ E vII(S) is bounded by qJ(O). The set vIIl(S) of functions JI(S) such that qJ(O) = 1 is therefore a compact convex base for the

E

cone vII(S).

6.5. Theorem. The cone vII(S) is an extreme subset of g>b(S) and Jll(S) is a Bauer simplex with ex(vIIl(S)) = S+ . For qJl' qJ2 E vII(S) also qJl . qJ2 E vII(S). A function qJ E g>b(S) is completely monotone if and only if the representing measure J.1 is concentrated on S+ . The previous theorem shows that A(S) ~ g>b(S). If qJ = qJl + JI(S) and qJi E g>b(S) has representing measure J.1i E M +(S), i = 1, 2, then J.1 = J.1l + J.12' where J.1 E M +(S +) is the representing measure for qJ. It PROOF.

qJ2

E

132

4. Main Results on Positive and Negative Definite Functions on Semigroups

follows that J.11' J.12 are concentrated on S+ so qJl' qJ2 E ..4t(S), and we have shown that vII(S) is an extreme subset of g>b(S). By transitivity of extremality an extreme point of ..4t 1(S) is also an extreme point of g>~ (S), hence ex(vII I(S)) ~ S +, and in fact there is equality since S+ ~vlll(S)nex(g>~(S)). For J.1, vEM+(S+) we have supp(J.1*v)~S+, cf. 2.3, and it follows that ..4t(S) is stable under multiplication. By unicity of the representing measure for qJ E g>b(S) it follows that vII 1(S) is a simplex, and that the representing measure for qJ is concentrated on S + if qJ E g>b(S) is completely monotone. D In analogy with Theorem 6.4 it suffices to check the defining conditions for a completely alternating function for elements in a generator set.

6.6. Proposition. Let G be a generator set for S. A function t/J: S -+pletely alternating if and only if Vat ...

Vant/J(s)~O

for

SES

and

a 1, ... ,an EG,

~

is com-

n~1.

PROOF. Suppose the conditions of the proposition hold. For s E S\ {O} there exist aI' ... , an E G with s = a 1 + ... + an, and the identity (1) in the proof of Lemma 5.2 gives in this case

I - Es =

n

n

L

E~f(I

(1"*1 j==1

- Eaj )I-(1j.

Applying the above procedure to elements s l '

(I - ESt) ... (I - ESk)t/J = VSt hence

... , Sk •••

E S\ {O} we find

VSkt/J ~ 0,

D

t/J E deS).

6.7. Theorem. The cone .s:I(S) is an extreme subset of JVI(S). A function t/J E JV'(S) is completely alternating if and only if the Levy measure J.1 is concentrated on S+ \ {I}. PROOF. If t/J E deS) we have by 6.3 and 6.5 that L\a t/J E g>b(S) for a E S, hence t/J E JVI(S) by Theorem 3.20. If t/J E deS) and t/J = t/Jl + t/J2' where t/Jl' t/J2 E JV'(S), then L\a t/J = L\a t/J 1 + L\a t/J2 E ..4t(S) and L\a t/J l' L\a t/J2 E g>b(S) for a E S. As vII(S) is extreme in g>b(S) we get L\a t/J l' L\at/J2 E vII(S), and by Lemma 6.3 we conclude that t/Jl' t/J2 E deS), and have thereby shown that

deS) is extreme in JV'(S). Let t/J E JV'(S) have the Levy-Khinchin representation t/J(s}

= t/J(O} + q(s} +

f.

JS

(1 - p(s)} dJl.(p}. \{I}

It is easy to see that t/J(O), q E .s:I(S) and that 1 - P E deS) when pES +\ {I}, so if J.1 is concentrated on S +\ {I} then t/J E deS). Let us next suppose that t/J E deS). Since deS) is extreme in JV'(S) we conclude that

t/Jis}:=

i

Ta

(1 - p(s)} dJl.(p}

~6.

133

Completely Monotone and Alternating Functions

belongs to .s:#(S), too, where Ta = {p ~at/Ja E Jt(S), hence

o ~ dat/Jia) =

i

E

SIpea)
b(S) and deS) = JVl(S). PROOF. If Sis 2-divisible every semicharacter is nonnegative, so the assertions follow from 6.5 and 6.7. D

6.9. Remarks. (1) It can happen that vII(S) = g>b(S) without S being 2divisible. Let S = {O, a, b} be the commutative semigroup with neutral element 0 and a + a = a + b = b + b = a. Then S is not 2-divisible, but S* = S has two elements Po == 1, Pl(O) = 1, Plea) = Pl(b) = 0 so vII(S) = q>b(S) = g>(S). (2) Suppose G is an abelian group considered as a semigroup with the identical involution. Then G is the group of homomorphisms of G into the multiplicative group { - 1, I}. In this case vII(G) = g>b(G) if and only if G is 2-divisible. In fact, if G is not 2-divisible then G/G 2 has at least two elements, where G2 = {2glg E G}, and every element s E G/G 2 satisfies 2s = o. If s E G/G 2 , S t 0, then (T) for all pES 0

for a triple (T, S, v) as described above will play some role for a converse of Theorem 1.2.

1.4. Definition. A triple (T, S, v), where T and S are abelian *-semigroups, and v: T -+ S is a mapping fulfilling the conditions (S 1 )-(S4)' will be called a Schoenberg triple. 1.5. Theorem. Let (T, S, v) be a Schoenberg triple. If'thefilnction qJ: S -+ C is bounded and positive definite, then qJ Vn is positive definite on Tn for every nE N. 0

In view of Theorem 4.2.8 it is enough to show that p all pES. This, in fact, is true because PROOF.

p vit 0

I' ... ,

tn ) =

0

Vn E

r!J>(T n ) for

pet V(t ») = }]/(V(t)) j

l

n

=

TI p

0

vonit l '

... ,

t n)

j= 1

(where n j denotes the canonical projection) is a product of positive definite D functions. The restricted dual of the semigroup (~+, +) is given by the exponentials A ~ 00, as we have seen earlier. This shows that a triple (T, S, v) where S = ~+, is a Schoenberg triple if and only if v is a nonnegative negative definite function on T for which (Sl) and (S3) hold. In particular, if T is a connected topological *-semigroup and v: T -+ [0, oo[ is a continuous negative definite function satisfying v(O) = 0 and v $ 0, then (T, ~+ , v) is a Schoenberg triple. sHe-AS, 0 ~

1.6. Example. The space IP of all real sequences whose pth power is summable is a Banach space only for p ~ 1, i.e. the function IIxllp:= IxjIP]l/P is a norm only for p ~ 1. Nevertheless IP as well as II· lip is well defined also for

[I

~ 1.

149

Schoenberg Triples

o< p
r(B) = {q>:

[R + -+ [R I q>

continuous, q>( 11·11) E &>(B)},

where positive definiteness refers to the group sense. The determination in general of &>r(B) seems to be open. If dim B = 00 then it is a subset of {2 J1( t 2 ) IJ1 E M~ ([R + ) }, and the sets are precisely equal when B is a Hilbert space. The following result of Einhorn (1969) shows that &>r(B) may be degenerate.

154

5. Schoenberg-Type Results for Positive and Negative Definite Functions

2.3. Theorem. Let qJ: [R+ ~ [R have the property that qJ(llxll) is positive definite on the Banach space C[O, 1]. For some rJ. E [R + we then have qJ 1]0, 00 [ == rJ. and qJ(O) ~ rJ.. In particular, .9r(C[O, 1]) consists only oj'the nonnegative constants. PROOF. For n ~ 2 let {(i,j) 11 ~ i

If 0 < s

~ t ~

<j

~

n} be the disjoint union of M and N.

2s we define (i,j) d(i, j) :=

{::

E

M,

(i,j)EN,

and d(j, i) := d(i,j) for i < j as well as dei, i):= O. The finite set {I, ... , n} together with d is a metric space and can hence be embedded in C[O, 1] by a famous theorem of Banach and Mazur (cf. Banach 1932, p. 169). This means that we can find {x b . . . ,xn } £; C[O, 1] such that (i,j)EM, (i,j) EN.

By assumption we have

o~

n

L cicjqJ(IIX i - Xjll) = [qJ(O) - qJ(t)] . L cf

+ qJ(t)(L Ci)2

i,j= I

+ 2[qJ(s)

- qJ(t)] . L CiCj

(2)

M

for all (C l , . . . , Cn) E [Rn. If M = 0 and L Ci = 0 this shows qJ(O) ~ qJ(t). Choosing n = 2m to be even, C 1 = ... = Cm = 1, cm + I = ... = cn = -1 and M := {(i,j)1 i < j, cic j = -I}, inequality (2) reduces to [qJ(O) - qJ(t)] . 2m ~ 2[qJ(s) - (p(t)] . m2 ,

and since this holds for all mEN, we see qJ(S) ~ qJ(t); analogously, the complementary choice of M gives qJ(s) ~ qJ(t). Finally, if all Ci = 1, (2) implies 0 ~ [qJ(O) - qJ(t)] . n + qJ(t) . n2 for all n, hence qJ(t) ~ O. D It seems to be open again, if .9r(B) can be degenerate also for some finite dimensional Banach space B. A closely related question was raised by Schoenberg (1938b): When is it true that exp( -llxlI1') is positive definite on a Banach space for some y > 0; if this is the case, which value of y then is the maximal possible? By homogeneity of the norm exp( - I x I 1') is positive definite if and only if I x I l' is negative definite, so if we put y*(B):= sup{y ~

0lllxll Y is negative definite on B},

Schoenberg's problem is equivalent to the determination of y*(B). By 3.3.3 we always have y*(B) ~ 2 and Lemma 2.2 shows that y*(B) = 2 is characteristic for Hilbert spaces. Schoenberg was particularly interested in

155

§3. Functions Operating on Positive Definite Matrices

the maximum norm on

~n

and stated the following explicit formula for

X,yE ~:

exp[ -(Ixl v Iyl)]

= n22

foo foo_ooexp[i(xu + yv)] -00

du dv x [1

+ (u + v)2][1 + (u

_ V)2]'

showing the negative definiteness of the maximum norm on ~2. In fact, as Herz (1963b) has shown, each norm on a two-dimensional real vector space is negative definite, but the maximum norm on ~3 is not negative definite. Hence, y*(~3, "·1100) < 1; already Schoenberg found that y*(~4, II

·1100)
O. The second statement may be proved similarly. D We now introduce C 1 to be the set of all functions qJ: [ -1, 1] --+ IR which operate on positive definite matrices in the sense that whenever (a jk) is a positive definite matrix with entries ajk E [ -1, 1], then (qJ(a jk » is again positive definite. By Schur's theorem (3.1.12) C 1 is a multiplicative closed convex cone in IR[-I, ll. It is immediate that each qJ E C 1 is nonnegative on

[0, 1]. For t

E [

-1, 1] the matrix

G ~)

qJ( 1) ( qJ(t)

if qJE C 1, and so IqJ(t) I ~ qJ(l) for all t K 1 :=

is positive definite, hence so is

qJ( t) )

qJ(l) E

[-1,1]. Therefore

I

{qJ Eel qJ( 1) =

I}

is a compact convex base for the cone C l '

3.3. Lemma. Let qJ E C 1 and t E [-1,1]. Put qJl(S):= qJ(s) + qJ(st), qJ2(S):= qJ(s) - qJ(st) }or s E [ -1, 1]. Then qJl and qJ2 also belong to C l' PROOF. Let (ajk) be a fixed positive definite n x n matrix, where ajk E [ -1, 1] for allj, k = 1, ... , n, and let (c 1 , • •• ,cn ) E IRn also be fixed. Define : [ -1, 1] --+ [R by n

(s):=

L

j, k= 1

cjc k qJ(sajk)'

We claim that E C 1. In fact, for any positive definite m x m matrix (b pq ), Ibpql ~ 1 for all p, q = 1, ... , m, and for all (d 1, ... , dm ) E IRm we get m

L

p,q=1

m

dpdq(b pq)

=

L

n

L

p,q=1 j,k=1

dpdqcjCkqJ(bpqajk)'

158

5. Schoenberg-Type Results for Positive and Negative Definite Functions

which is nonnegative because the tensor product (ajk) ® (b pq ) is positive definite, too, see 3.1.13. In particular, n

I(t) I ~

L

(1) =

for all

CjC k qJ(a jk)

t E [ -1, 1]

j, k= 1

D

and this shows that qJ l' qJ2 both belong to C l'

3.4. Theorem. The set K 1 is a Bauer simplex whose extreme points consist oj' the monomials t ~ tn, n E No, and the two discontinuousjilnctions 1{1} - 1{-1} and 1{- 1 , 1} • Let qJ E ex(K 1) and t A(t) ~ 0 such that

PROOF.

A

=

qJ(s)

hence A = 1

-1, 1]. By Lemma 3.3 there exists a constant

E [

+ qJ(st) =

for all

AqJ(S)

s E [ -1, 1],

+ qJ(t) and therefore qJ(st)

=

qJ(s)qJ(t)

for all

s,

tE [

-1, 1],

i.e. qJ is multiplicative. It is well known that there exists lJ. E [0, 00] such that for all s E ]0, 1], qJ(s) = sa where 100 := 1, Soo := 0 for s E ]0, 1[. If lJ. E [R+ \N o then by Lemma 3.1 there is a positive definite matrix (a jk ) with ajk E]O, 1] such that (aJk) is not positive definite. We conclude that lJ. E No u {oo}. Of course, qJ(O) E {O, I} and qJ(O) = 1 only if qJ == 1, hence for lJ. = 00 we get qJ = 1{-1, I} or qJ = 1{1} - 1{-1}' Let lJ. = n E No be finite. If qJ( - 1 ) = 1 then qJ(t) = I tin; by Lemma 3.2 n has to be even, therefore qJ(t) = t n for all t E [ -1, 1], except for the special case that n = 0 and qJ = 1[-1, 1]\{O}' If qJ( - 1) = -1 then qJ(t) = Itin. sgn(t) and for n > 0 we get from Lemma 3.2 that n must be odd, i.e. qJ(t) = t n for all t E [ -1, 1] holds again. Two cases are left: qJ(t)

=

sgn(t)

and

qJ(t)

= sgn 2 (t) =

1[-1,

1]\{O}(t).

We shall show that these two functions do not belong to K l' Let t 1 = 0, t 2 = n12, t 3 = n and put a jk :=! + ! cos(t j - tk),j, k = 1,2,3. The matrix

is positive definite, but

has not this property, because (1, -1, l)A(l, -1, 1)' = -1 < O. This finishes the first half of the proof.

159

§3, Functions Operating on Positive Definite Matrices

Let E:= {t H t n I n E No} u {1{1} - 1{-1}, 1{l, -I}}. By Theorem 3.1.12 the monomials belong to K 1 and so do their two limit points, hence E ~ K 1. Let S denote the multiplicative semigroup [ -1, 1], then K 1 ~ g>~(S) and E ~ S. In Theorem 4.2.8 we have seen S = ex(g>~(S)), so that ex(K 1) ~ E ~ KIn ex(g>~ (S))

and finally ex(K 1) = E. It is very easy to see that every function in K 1 has a unique integral (= series) representation over E. Therefore K 1 is a Bauer simplex. D

3.5. Corollary. Every qJ E C 1 admits a unique series representation oj' the form 00

qJ(t) =

L antn + a_ 1[1{1}(t)

- 1{_l}(t)]

+ a-2 1{-1, l}(t),

n=O

where all an ~ 0 and L an < 00. The jilnction qJ is continuous if' and only if' a_ 1 = a_ 2 = 0, and qJ is then even an analytic junction. Now let H denote an infinite dimensional real Hilbert space with scalar product and denote by X := {x E H IllxlI = I} the unit sphere of H. Restricted to X x X, the scalar product is a mapping onto the interval [-1,1], and plainly for every qJEC l the composed map qJ«x,y») is a positive definite kernel on X x X. The next theorem shows that the converse of this result also holds.

Ojor all t E [ -1, 1] and t/J := -log p,julfils the three conditions: (i) t/J(I) = 0; (ii) t/J is continuous; (iii) for each positive definite matrix (a jk) with entries in [ -1, 1] the matrix (t/J(a jk)) is negative definite.

If the conditions are fulfilled there exist uniquely determined numbers bb b 2 , ••• ~ 0, L bn < 00, such that 00

t/J(s) = L bn(1 - sn) n=l

jor aII

sE

[ -

1, 1].

PROOF. If P, > 0 and t/J := -log p, fulfils (i)-(iii) then t/J is a continuous function in C 2 , hence by 3.2.2 exp( -tt/J) is a continuous function in K 1 , i.e. a generating function for all t > O. In particular

p, = [exp( - t/J In)]n for all n E N, showing that /1 is infinitely divisible. Now let us assume /1 = (/1n)*n for suitable probability measures /11' /12' ... on No; equivalently, p, = (p'n)n for the corresponding generating functions. If P,(t o) = 0 for some to E] -1, 1[, then all derivatives of fi in to would vanish and then p, == 0 in contradiction to P,(I) = 1 and to {l's continuity on [-1, 1]. Hence fi(t) > 0 at least for all t E ] -1, 1] and consequently all the functions {In are strictly positive on ] -1, 1]. Now t/J:= -log p, is well defined, continuous on [ -1, 1] and finite on ] -1, 1], but still we have to exclude that t/J( -1) = 00. Let (a jk ) be a positive definite matrix, ajk E] -1, 1] for allj, k. Then (t/J(ajk)) is negative definite, because for all n E N

(e-o/n)!/I(a jk ») = (p,(a jk)l/n) = (Pn(ajk)) is positive definite by Proposition 3.7. In particular, t/J I] -1, 1] is a nonnegative negative definite function on the semigroup ] -1, 1], therefore bounded by Proposition 4.3.4, and since t/J is continuous on [-1, 1] we get t/J( -1) < 00, i.e. P,( -1) > 0, so that t/J is indeed a (real-valued) continuous function in the cone C 2 • The asserted representation of t/J follows immediately from Corollary 3.12. D

3.15. Corollary. Let /1 E M~(No) be infinitely divisible. Then /1 is the distribution of a random variable X = L:= 1 nXn' where the Xl' X 2' ... are independent, and where each X n is Poisson distributed with parameter bn, and L:=l bn < 00. PROOF.

L bn
0; (iii) qJ2(t) = qJ(t 2) for all t E [ -1, 1].

§4. Schoenberg's Theorem for the Complex Hilbert Sphere In the preceding section we have seen that a continuous function qJ: [ -1, 1] such that qJ( y») is positive definite on the unit sphere of some infinite dimensional real Hilbert space, has a power series representation qJ(s) = ans n with an ~ 0 and an < 00. We are now going to extend this result to the case of a complex Hilbert space H (always infinite dimensional), and of course we have to replace the interval [ -1, 1] by the closed unit disc D:;:::: {z E Clizi ~ I} in the complex plane. Let lr := {z E C/izi = I} denote the torus group. When we consider lr as a discrete group, we shall write lr d instead of lr. In analogy with the real case we introduce C to be the set of all functions qJ: D ~ C operating on positive definite matric'"3 in the sense that (qJ(ajk)) is positive definite whenever (a jk ) is a positive definite matrix with entries in D. By the Schur Product Theorem (3.1.12) C is a multiplicative closed convex cone in CD. It is immediately seen that each qJ E C is nonnegative on [0, 1].

<x,

~ [R,

L

L

167

§4. Schoenberg's Theorem for the Complex Hilbert Sphere

G;)

For zED the matrix

is positive definite, hence so is

qJ(l) ( qJ(z) if qJ E C, and therefore qJ(z) showing, in particular, that

qJ(Z») qJ(l)

= qJ(z) as well as IqJ(z) I ~ qJ(l) for all zED,

K := {qJ E C IqJ( I) = I} is a compact convex base for C. For the proof of Theorem 4.5 below, in which the extreme points of K are determined, we need several lemmas which we prove first. 4.1. Lemma. The set of numbers c E C for which the matrix

MC:=(~eel ~ ~) is positive definite is the" pear-like" subset Q

n of the unit disc (f"igure 2)

= {cEDI[3 - 2 Re(c)JlcI 2

~ I},

having the following properties: (i) {cECllcl ~!} ~ Q. (ii) Q is symmetric about the x-axis and its boundary curve in the upper half-

plane is y

with y'(l) =

=

y(x)

= (1-

f2X+1

x)V~

-J3.

(iii) For t E [!, IJ and

for

eE [ -1t, 1tJ we have c = te lei

iO

x

E [

E Q if and only if

~ e(t):= arccos(~t-l - !t- 3).

y

y(x)

1 x

-1

Figure 2

-!, IJ

168

5. Schoenberg-Type Results for Positive and Negative Definite Functions

PROOF.

We have detG

~) =

1-

Jel 2 and

det Me = 1 - [3 - 2 Re(c)]lcI 2 so the description of Q follows from Theorem 3.1.17. (i) is clear. (ii) If c = x + iy E Q then x ~ O,hence

E [

-1, 1] and det Me = 1 - (3 - 2x)(x 2 + y2)

2

y~

(1 - x)2(2x

3-2x

+

1)

'

which shows that x E [ -!, 1] and

IYI ~

(1 -

fiX+l x\j3=2X'

and the converse is also true. By differentiation of the function y = y(x) we get y'(I) = -)3. (iii) For c = te U) we have det Me = 1 - [3 - 2t cos 0]t 2 ~ 0 if and only if cos 0 ~ ~t-l - !t- 3. Since t~~t-l - !t- 3 maps [!, 1] onto [-1,1] the assertion follows. D

4.2. Lemma. Let qJ(tei(J) = tkeip(J, where k E r\Jo, P E 71... If' qJ E K then Ip I ~ k. If qJ E K we get by Lemma 4.1 that qJ(Q) ~ Q, in particular qJ(tei(J(t» = tkeip(J(t) E Q for! < t < 1, k where O(t) = arccos(~t-l - !t- 3). If t > ! and IpO(t) I < 1t, which is true for t sufficiently close to 1, we have IpO(t) I ~ O(t k), hence · O(tk) 1· kt k- 10'(tk) k I m - - = 1m = Ip I -~ 1t-l O(t) t-l O'(t) by I'Hopital's rule because PROOF.

lim O'(t) = t-l

-)3.

The direct evaluation of this limit is tedious but elementary. It is, however, no coincidence that this limit is y'(I) because

lim ~ (te i6 (1» = 1 + i lim (}'(t) t-l dt t-l is a tangent vector to y = y(x) at x = 1.

D

4.3. Lemma. Let qJ E C and ZED and define thefunctions qJl' qJ2' qJ3' qJ4: D -+ C by

qJl, 2(W) = qJ(w) qJ3,4(W) = qJ(w)

± ![qJ(zw) + qJ(zw)] ± (1/2i)[qJ(zw) - qJ(zw)].

Then qJl' ... , qJ4 all belong to C, too.

~4,

169

Schoenberg's Theorem for the Complex Hilbert Sphere

Let (ajk) be a fixed positive definite n x n matrix with entries in D, and let (c l' . . . , Cn) E C n also be fixed. Define the function : D -+ C by

PROOF.

n

L CjCk ({J(zajk)·

(z):=

j, k= 1

We claim that E C. In fact, for any positive definite m x m matrix (b pq ) with entries in D and for all (db .. . , dm ) E cm we have m

L

p, q = 1

m

n

L

dpdq(b pq) =

L

p, q = 1 j, k = 1

dpcjdqCk ((J(bpqajk) ~ 0

because the tensor product (b pq ) ® (ajk) is positive definite by Corollary 3.1.13. Therefore n

I(z) I ~

L CjCk ((J(a jk ),

(1) =

j,k= 1

hence, denoting G = Re( 0

[x, y]-t = e-t1og[x,y] = (x oYo)-t(l - cp(x, y»-t. The right-hand side is positive definite by Corollary 3.1.14 so that t/f(x, y) := log[x, y] is a negative definite nonnegative kernel on X x X. The equation

174 cosh d

5. Schoenberg-Type Results for Positive and Negative Definite Functions

= exp(t/J) has for d, t/J

~

0 the unique solution

+ 10g(1 + Jl

d = t/J

- e- 2 t/1),

and therefore d is negative definite, too, by Corollary 3.2.10. Now we can state the above-mentioned result of Faraut and Harzallah (1974).

5.1. Theorem. Letj': [R+

~

[R be an arbitrary given junction. Then

f(log cosh d(x, y» is positive definite on X x X if and only iffE gpb([R+). The kernel f(log cosh d(x, y» is negative definite if and only iff E %'( [R +). PROOF. IffE gpb([R+), thenf(s) and then

=

flO,

00] e-J.s

dJ.l()") for some J.l EM +([0,00])

f(log cosh d(x, y» = f(log[x, y]) =

f

e-. (ii)" Suppose ajXs j = 0 on S*, i.e.

PROOF.

I

S l'

... ,

Sn E

S are distinct points and that

n

I

ajp(Sj)

=

pES*.

for

0

j= 1

In order to prove that a 1 = 0 we choose Pj E s* for j Pj(Sl) =f Pj(Sj), which is possible by (i), and define

=

2, ... , n such that

n

j'(s)

Il (Pj(s) -

=

Pj(Sj»(Pj(Sl) - Pj(Sj»-l,

SE

S.

j= 2

Thenj'belongs to the linear span of S* in CS andf(sl) j = 2, ... ,n, hence

= 1,f(sj) =

0 for

n

a1 =

I

ajf(sj) = O.

j= 1

D

The implication" (ii) => (i)" is trivial.

1.9. Suppose that the equivalent conditions of Proposition 1.8 are verified. Then C S and V(C) form a dual pair under the bilinear mapping

\ cP,

Jl

aj xSj )

=

Jl

ajcp(s),

(5)

which is well-defined because of 1.8(ii). The topology of pointwise convergence on C S is compatible with this duality. In fact, if L: C S -.. C is a continuous linear functional there exist a finite set D £; Sand c; > 0 such that IL((S) = Jf(S) follows from Theorem 1.10. D


(S) and also where Jf(S) =t= &>(S).

1.12. Exercise (The Method of Moments). Let S be a finitely generated abelian semigroup with involution and let (Iln) be a sequence from E + (S*) with corresponding sequence of moment functions (qJn)' Prove that if qJ E Jf(S) is a determinate moment function with representing measure Il E E +(S*) and if lim n oo qJn(s) = qJ(s) for each s E S then lim n oo Iln = Il weakly. -+

-+

§2. The One-Dimensional Moment Problem In this section, we will apply the results of §1 to the semigroup S = (No, + ), the involution being the identical. For x E [R the mapping Px: No -.. [R given by px(n) = x n , n ~ 0, is an element of N6 and it is easy to see that the mapping x H Px is a topological semigroup isomorphism of (IR, .) onto N 6. The functions Xn' n E No, cf. (3) in §1, will be identified with the monomials x H x n, which are linearly independent, so V([R) is the vector space of polynomials in one variable with real coefficients. The following lemma shows that the cone I: (see (6) in §1) is equal to V(IR) + .

186

6. Positive Definite Functions and Moment Functions

°

2.1. Lemma. Let P be a polynomial with real coefficients and suppose p(x) ~ jor x E [R. Then there exist two polynomials with real coefficients PI' P2 such that p(x) = PI (X)2 + P2(X)2 jor x E [R. PROOF. Complex roots of P appear in conjugate pairs and real roots have an even multiplicity. Therefore we may write p(x) = r(x)r(x), x E [R for some polynomial r with complex coefficients. Writing r = PI + iP2' where PI' P2 are real polynomials, we get p(x) = PI (X)2 + P2(X)2 for x E [R. 0

By Theorem 1.11 it follows that &>(~o) = Jr(~o), so we have proved the following theorem of Hamburger, called the solution of Hamburger's moment problem, cf. Akhiezer (1965).

2.2. Theorem. A sequence s = (sn)n~o oj'real numbers is the sequence oj' moments oj' a measure /l E E + ([R) where E+(IR)

=

{JlE M +(IR)I flxnl dJl(x) < 00 for all n ~ o}

i.e. is of the form Sn =

f

x n dJl(x),

n

(1)

= 0, 1, ... ,

if'and only if's E &>(~o). The set E+([R, s) oj'measures /l E E+([R) satisjying (1) is a nonempty compact convex set in M +([R) in the vague topology. 2.3. Remark. A sequence s = (sn)n~O belongs to &>(~o) if and only if for every n ~ the matrix ((Sj+k)o~j,k~n) is positive definite. Using Theorem 3.1.16 it follows that s is strictly positive definite if and only if

°

n = 0, 1, 2, ....

for Let SE&>(~o) and /lEE+([R, s). For {co, c l , k L~=o CkX and have

j.~/jCkSj+k =

f

... ,

p(X)2

cn}

~

IR we define p(x)

=

dJl(x),

so s is strictly positive definite precisely when supp(/l) is an infinite set.

2.4. Let F be a closed subset of [R. The F -moment problem consists in characterizing the F -moment sequences, i.e. the real sequences s = (sn)n ~ 0 of the furm Sn

=

L

n

x dJl(x),

n

~

where /l E E + ([R) is supported by the closed set F.

0,

187

§2. The One-Dimensional Moment Problem

Hamburger's moment problem corresponds to F

°

= IR. The case of

F = [ -a, a], a ~ has been solved in Proposition 4.4.9, and in the special case a = 1 which we will call Haviland's moment problem, two different characterizations of [-1, 1]-moment sequences s have been found, (a) SE&,b(N o), (b) sE&'(N o) and s - E 2 sE&,(N o), the latter going back to Haviland (1936), cf. 1.2. The case of F = [0, 1] is Hausdorff's moment problem and the [0, 1]-moment sequences are precisely the completely monotone functions on the semigroup (No, +), see 4.6.11. We shall now consider the case F = [0, oo[ which is called Stieltjes' moment problem, solved by Stieltjes in 1894. The [0, 00 [-moment sequences are also called Stieltjes moment sequences.

2.5. Theorem. For a sequence S tions are equivalent:

=

(sn)n

~0

of'real numbers the following condi-

(i) s, E 1 SE &'(N o). (ii) t = (So, 0, Sl' 0, S2'· ..) E &'(N o). (iii) There exists J1 EM +([0, oo[) such that Sn

PROOF.

"(i) ~ (ii)" Let

tn

=

= sn/2

f"

n

n

x d/l(x),

for n even and t n

~

=

0.

°

for n odd. For

we then have n

I

n

n CjCktj+k

=

j,k= 0

I

CjCktj+k

+ L

[(n - 1 )/2]

[nI2]

=

I

CjCkt j + k

j,k= 0 j, k odd

j,k=O j, k even

C2pC2qSp+q

I

+

p,q=O

C 2p + 1 C 2q + 1 Sp+q+ 1

p,q=O

showing that t E 9(N o). "(ii) ~ (iii)" By Hamburger's theorem (2.2) there is t 2n + 1

=

f

x 2n + 1 d(J(x)

(J

E

~

°

E + (IR) such that

= 0,

n

~

0,

hence

n

~

0,

where J1 is the image measure of (J under the continuous mapping x ~ x 2 of IR into [0, 00 [. "(iii) ~ (i)" is clear since E 1 S is represented by the positive Radon measure xJ1. D

188

6, Positive Definite Functions and Moment Functions

In analogy with Theorem 2.2 we can give an integral representation of all functions ljJ E JV(N o) with a representing measure which is not necessarily unique.

2.6. Theorem. Ajilnction ljJ: No ~ lR is negative definite if'and only if'it has a representation oj' the form ljJ(n)

where a,

r

= a + bn - cn 2 + bE

(1 - x n

n(1 - x)) d/l(x),

-

J~\{l}

lR, c ~ 0 and Il E M +(lR\ {I}) satisfies

f.

(1 - X)2 dll(X)
0 such that supp(a l/J £; lR\] - 8,8[ for t E ]t o - 8, to + 8[, and then

x n E CO(lR\] -

8,

e[)

whenn ~ -1. We shall finally prove that qJt E 9(7L.) is determinate. We first establish that (1jx 2 )a l/l generates a determinate Hamburger moment sequence, and for

201

§4. The Two-Sided Moment Problem

this it suffices to see that

inf{f p(X)2

:2

= I} = 0,

dO"l/t(x)!p polynomial, p(O)

(5)

cf. Akhiezer (1965, pp. 49 and 60). Now, l/x E 2 2(0'1/t) and the set of polynomials is dense in 2 2(0' lit) because a 1/t is Nevanlinna extremal, so for e > 0 there is a polynomial q such that e>

f(~ -

q(x)

r

dO" l/t(X)

=

f

(l - xq(X»2 :2 dO" l/t(X),

and (5) follows because p(x) = 1 - xq(x) is a polynomial verifying p(O) = 1. To see that CP, is determinate we consider Jl E E + (lR\ {O}, CPt) and will prove that Jl = a Ill' By assumption

r

J~\{O}

x n d/1(x) =

r

J~\{O}

x n dO" l/t(X)

for

n E"Z,

in particular for

n

~

and we conclude that (1/x 2)Jl = (1/X 2)0'1/n whence Jl = a lit.

0,

D

4.6. Example. We will give an example of an indeterminate two-sided moment sequence. The log-normal distribution has the following density function on ]0, 00 [

which generates cP E 9("Z) given as q>(k)

= f'xo_1_xk-le-O/2)(IOgx)2dx,

Jo

kE"Z.

Jhc

Substituting t = log x yields cp(k) = _1_

Jhc

foo

e-O/2)t2+kt dt = e(/2 )k 2 _1_

Jhc

-00

foo

e- O/2 )(t-k)2 dt = e(1/ 2)k 2.

-00

(This should be compared with the fact that ljJ(k) = - k 2 is negative definite on ("Z, +) in the semigroup sense.) That cP is indeterminate follows from the fact that all the density functions da(x) = d(x)(l

+ a sin(2n

log x)),

XE

]0,

00[,

202

6. Positive Definite Functions and Moment Functions

where a E [ -1, 1], generate the same moment sequence

rooxk sin(2n log x) d(x) dx = _1_ Jo ~

= e(l/2)k qJ E lR~

4.7. Exercise. A function

foo

2

e-(1/2)t +kt

qJ

because for k E 7L

sin(2nt) dt

-00

2

_1_

~

foo

e-(1/2)y2

sin(2ny) dy

=

O.

-00

is called a two-sided Stieltjes moment 00 [) such that

sequence if there is a measure J1 E M + (]O,

cp(k) =

1 00

x k df.l(x)

for

k E 7L.

Show that the following conditions are equivalent for

qJ E lR~:

(i) qJ is a two-sided Stieltjes moment sequence. (ii) f: 7L -+ lR defined by

f(k) = {cp(~).

keven,

k odd,

0, (iii)

belongs to 9(7L). qJ, E 1 qJ E 9(7L).

4.8. Exercise. Let S = (7L k , +). Show that S* is isomorphic and homeomorphic with «lR\ {O})k, .) under the mapping x ~ Px where px(n) = x" = x = (Xl' ... ,xk) E (lR\ {O})k and n = (n l , ••. ,nk) E 7L k. Show that, in the terminology of §1, L is closed in V(lR) and a proper subset of V(lR)+ when k ~ 2. Conclude that there exist functions qJ E 9(7L k )\Yf(7L k ) when k ~ 2. X~l ••• X~k for

4.9. Exercise. Show that a function 1jJ: 7L semigroup sense on (7L,

r/J(n)

=

a

+ bn

-+

lR is negative definite in the representation of the form

+) if and only if it has a

- cn 2

+

r

J~\{O, l}

(1 - x n

n(1 - x)) df.l(x),

-

nEZ,

where a, b E lR, c ~ 0 and J1 E M + (lR\ {O, 1}) satisfies

r J

O (r) where a PROOF.

=

+

al{o}(r)

f:oo e

rx

dJ-l(x)

0, and J-l E M + (IR) is such that erx is J-l-integrable for each r E Q + . For x E IR the function px: 0+ -+> IR defined by px(r) = erx is a semi~

character and so is p_ 00 := l{o}. Conversely, if p E S* then p(l) ~ 0 and defining x = log p(l) E [R u {- oo} we have p = Px. It is easy to see that x ~ px is a topological semigroup isomorphism of ([ - 00, 00[, +) onto S*, the topology on [ - 00, oo[ being the obvious with {[ - 00, -n[ln E N} as a base for the filter of neighbourhoods of - 00. Let qJE(!jJ(O+) and assume cp(O) = 1. For each nEN we have that (qJ(k/n))k?:O is a Stieltjes moment sequence because

k~

and

qJ

k + 1) = qJ (k;; + 2 2n1) (-n-

are positive definite on (No, +), cf. Theorem 2.5. Therefore there exists J-ln E M~([O, oo[) such that

q>(~) = Loox The

mappingj~: [

-

k

00, 00 [ -+>

j~(x) =

[0,

00 [

ex / n

n

~

0,

n E N.

defined by

for

is a homeomorphism, so there exists (cf. 2.1.14), hence

qJ(~)

k

for

dJ-lix)

xE[-

Tn E

= [

00,

oo[

M~([ - 00, oo[) such that T~"

= J-ln

e(k/n)x dTn(X).

J[-oo.OO[

By 2.4.6 and 2.4.10 there is an increasing sequence nt < n2 < ... in N such that T(nk)! converges vaguely to a measure T E M +([ - 00, oo[) of total mass ~ 1. Let r = p/q E 0+ be fixed and assume that p, q E No have no common prime factors, q =t= O. For kEN such that nk ~ q we find k )!) qJ(r) -- qJ (r(n -( )' nk •

f

erx dT(nk)!(x),

[-00,00[

and using that for each nonnegative continuous functionj'on a locally compact space the integral j'dJ-l is lower semicontinuous in J-l with respect to the vague topology (cf. 2.4.1), we find

J

r

J[ - 00, oo[

erx dr(x)

~

q>(r),

210

6. Positive Definite Functions and Moment Functions

which shows that erx is r-integrable for all r E Q+. Let hE CC+([ - 00, oo[) satisfy 0 ~ h ~ 1, hex) = 1 for x E [ - 00, 0], hex) = 0 for x ~ 1. Then cp(r) =

r

J[ -

+

erxh(x) dr(nk)!(x) 00,

oo[

r

J[ -

e(r+ l)x(l - h(x»e - x dr(nk)!(x), 00,

oo[

(6)

and since erxh(x) has compact support in [ -

f J[ Furthermore, (1 - h(x))e- X

E

00,

oo[ the first term converges to

erxh(x) dr(x). 00,

oo[

CO([ -

00,

oo[) and the sequence

converges vaguely to e(r+ l)x r with the total masses bounded by cp(r + 1). By Proposition 2.4.4 it follows that the second term of (6) converges to

r

J[-

e(r+ l)x(l - h(x»e- X dr(x) 00.

=

oo[

r

J[ -

erx(l - h(x» dr(x), 00,

oo[

hence cp(r) showing that

=

r

J[ -

qJ E Jff( Q +).

erx dr(x) 00,

oo[

Putting

cp(r)

Q

=

r({ -

= al{o}(r) +

00 })

and J-l

= r I~ we have

{'' oo erx d/-l(x),

hence lim cp(r) =

J-l(~)

= cp(O) -

Q.

r-+O

The function

is well defined and continuous on the closed half-plane {z Eel Re z ~ O} and holomorphic in its interior, hence uniquely determined by cp. On the imaginary axis is the Fourier transform of J-l which shows that J-l is uniquely D determined by cp. By Proposition 5.6 and Theorem 5.4 we get that (Q~, +) is a perfect semigroup for k ~ 1. For x = (Xl' ••• , x k ) E [ - 00, OO[k and r = (r 1 , ••• , rk ) E (p~ we define <x, r) = 1 xjr j and px(r) = e<x,r) and so we have:

LY=

211

§5. Perfect Semigroups

5.7. Proposition. For every qJ E .?J>(O~) there is a unique measure J-l E M + ([ - 00, 00 [k) such that

qJ(r) =

i

e<x, r) dJ-l(x)

r E O~.

jor

[- 00, oo[k

As an application of this result we prove:

5.8. Theorem. A necessary and sufficient condition that a jilnction qJ: IR~

-+>

IR

can be represented as q>(y)

=

r e<x,y) dJ1(x)

for

J~k

y

E

IR~

(7)

with J-l E M + (IRk), is that qJ is continuous and positive definite on (IR~, +). The measure J-l is uniquely determined by qJ. PROOF. It is clear that qJ given by (7) is continuous on IR~ and positive definite on (IR~, +). Conversely, if qJ has these properties the restriction of qJ to O~ belongs to [!J(O~) and is hence of the form

qJ(r) =

i

e<x, r) dJ-l(x)

for

rE

Q~

[- 00, oo[k

with J-l E M +([ - 00, OO[k). As in the proof of Proposition 4.4.7 the continuity of qJ at (0, ... , 0) implies that J-l is concentrated on IRk. The two continuous functions qJ and y

~

r e<x,

J~k

y)

dJ1(x)

agree on O~ and (7) follows. The uniqueness follows from Proposition 5.7.

o

5.9. Remark. The function qJ in Theorem 5.8 has a unique continuous extension iP to {z E C k IRe zj ~ 0, j = 1, ... , k} which is holomorphic in the interior. It is given as iP(z) =

and iP(iy), y E

f

e<x,z) dJ-l(X) IRk IRk, is the Fourier transform of J-l.

The results about 0+ can be extended to (0, +) with the identical involution. 5.10. Proposition. The semigroup (0, +) with the identical involution is perject. Every qJ E [!J(Q) has a unique representation as

jor where J-l E M + (IR).

r E Q,

212

6. Positive Definite Functions and Moment Functions

Since h: 0 + x Q + ---. Q defined by h(s, t) = s - t is a homomorphism and onto, it follows from 5.4, 5.5 and 5.6 that (0, +) is perfect, too. Furthermore, the proof of Theorem 5.5 shows that Q* is isomorphic (and homeomorphic) to h*(Q*). Noting that (Q~)* ~ ([ - 00, 00[2, +), cf. Proposition 5.7, we get for r E Q*, h*(r) = (x, y) E [ - 00,00[2, and s, t E 0+ PROOF.

e

XS

+ yr

= (h*( r))(s, t) = r(h(s, t)) = r(s - t),

implying x > - 00, Y > - 00 and y = -x, i.e. r(s) = eXs • Hence 0* is isomorphic with (IR, +) and a given qJ E &>(0) has the unique integral representation for where J1

E

r E Q,

M~(IR).

D

Also (Qk, +) is perfect and in analogy with 5.8 we find:

5.11. Theorem. A necessary and sufficient condition that afunction qJ: IRk ~ IR can be represented as cp(y)

=

r

J~k

e(x,y)

dJ-l(x)

with J1 E M + (IRk) is that qJ is continuous and positive definite in the semigroup sense on (IRk, +). The measure J1 is uniquely determined and qJ can be extended to an entire holomorphic jilnction on C k • Theorems 5.8 and 5.11 can be generalized to a non-semigroup setting, cf., Widder (1941, p. 273) and Akhiezer (1965, pp. 211 and 229).

5.12. Theorem. Let H £; IRk be an interval. A necessary and sufficient condition that a jilnction qJ: H ---. IR can be represented as cp(y)

=

r

J~k

e(x,y)

dJ-l(x)

jor

yEH

with J1 E M + (IRk) is that qJ is continuous and the kernel (x, y) H qJ(x positive definite on (tH) x (tH).

+ y)

is

The proof will not be given here, but let us indicate how a proof may run. The result is first established for k = 1 using the truncated moment problem, cf. Akhiezer (1965), and then extended to k dimensions by the technique of Theorem 5.4. We will now give an integral representation of the cone %(Q +) of negative definite functions on the semigroup Q+ .

~5.

213

Perfect Semigroups

5.13. Proposition. A jilnction l/J: 0 has a representation oj' the jorm

+ --+ ~

r

tjJ(r) = a + br - cr 2 + d1 O. Show that S is not perfect. 5.23. Exercise. Let qJ: IR --+ IR be a continuous function which is positive definite on the multiplicative semigroup (IR, .). Show that there are measures fJ. E M +(IR+), v E M +(]O, oo[) such that cp(s) =

r

J~+

1slY d,u(y)

+

r

1s I)' , sgn(s) dv(y),

s E IR.

J]o.oo[

5.24. Exercise. Show that (Q + , .) and (Q, .) are not perfect semigroups. Hint: Use that (Q+ \ {O}, .) is isomorphic with Z(oo).

Notes and Remarks In his fundamental paper" Recherches sur les fractions continues", Stieltjes formulated and solved the moment problem which bears his name. Later Hamburger generalized Stieltjes' result to moment sequences of measures on the whole real line. Moments of a measure have been studied, before Stieltjes, by Tchebycheff and others, but concerning the history of the moment problem we refer to Shohat and Tamarkin (1943). Our choice of the name "moment function" on a semigroup S is motivated by this classical theory, which corresponds to the semigroup (No, +). Likewise the symbol Yf(S) reflects the name of Hamburger. The results in 1.6-1.11 seem to be new but are, of course, known for some concrete semigroups. For a detailed ~tudy of Hamburger's moment problem, in particular of the set E + (IR, s), we refer the reader to the classical monographs by Akhiezer (1965) and Shohat and Tamarkin (1943). Results about denseness of the set of polynomials in 2 P(IR, fJ.) can be found in Berg and Christensen (1981,

223

Notes and Remarks

1983a). The F-moment problem in the case F = {x E IRlp(x) ~ O}, where p is a fixed polynomial, is studied in Berg and Maserick (1982). It contains a characterization of the polynomials p for which the set of {p ~ O}-moment sequences is equal to {sIs, p(E)s E &>(r\J o)}. Here N

(p(E)s)n =

L aksn+ k k=O

N

if p(x) =

L akxk. k=O

The F-moment problem where F = IR\U7= I ]a i , bi [ and a l < b l < a2 < b 2 < ... < an < bn is solved in Svecov (1939) for n = 1 and in Fil'stinskii (1964) for arbitrary n. Horn (1969b) showed that a nonzero positive definite sequence qJ: r\J o ~ IR is infinitely divisible if and only if t/J := -log qJ has the representation stated in Theorem 2.6. Already in 1888 Hilbert remarked that there exist nonnegative homogeneous polynomials in three variables which are not a sum of squares of homogeneous polynomials, cf. Hilbert (1888). His method is, however, rather involved and does not lead to a simple explicit example, cf. Gelfand and Vilenkin (1964, pp. 233-234) where Hilbert's construction is indicated. An example along these lines was found by Robinson (1969, 1973) and by Schmlidgen (1979). The polynomial xix~(xf + x~ - x~) + x~ obtained by making the polynomial of Lemma 3.1 homogeneous is probably as simple as possible since the degree must be at least 6, as shown by Hilbert. The paper by Motzkin (1967) contains a similar explicit example. A nonnegative homogeneous polynomial in two variables is a sum of two squares of homogeneous polynomials by Lemma 2.1. Theorem 3.4 is stated in Zarhina (1959) and also referred to in Gelfand and Vilenkin (1964, p. 235), but the proof seems incomplete. Zarhina states that it is unknown whether L(k) n A~k) is closed in A~k) and claims without proof that its closure is different from (A~k») + . Independently and almost simultaneously Theorems 3.2 and 3.4 were published by Berg et al. (1979) and Schmlidgen (1979). An explicit example of a function qJ E &>(r\J6)\~(r\J6) has been given by Friedrich in a paper to appear in 1984 (or later). Only few papers contain sufficient conditions for a function qJ E &>(Nt), k ~ 2, to be a moment function, cf. Devinatz (1957), Eskin (1960) and Nussbaum (1966). All these papers are based on results about the delicate problem of finding commuting self-adjoint extensions of symmetric operators which commute on a common dense domain in a Hilbert space. A survey of the operator theory approach to the moment problem is given by Fuglede (1983), cf. Exercise 3.15. The notion of Fmoment sequences and the .F-moment problem can be extended to k dimensions, and the first solution of the problem is due to Haviland (1935, 1936). His result is given in Exercise 3.16. For results in case of concrete sets F £; IRk, see Exercise 3.17 and McGregor (1980). If F is a compact convex set with nonvoid interior the problem is solved by Maserick (1977) and rediscovered by Cassier (1983). Semigroups of moments are considered in Berg (1984a), Buchwalter (1984) and Buchwalter and Cassier (1983a, b). In

224

6. Positive Definite Functions and Moment Functions

Petersen (1982) it is proved that qJ E Yf( N~) is determinate provided each of the one-dimensional moment sequences qJ(n 1 , 0, ... , 0), ... , qJ(O, ... , 0, nk ) is determinate. The results in 3.6-3.13 were summarized in Berg and Christensen (1983b). The two-sided moment problem has been studied by Jones et ale (1980, 1984). The first paper deals with the Stieltjes case where the representing measure is concentrated on JO, 00 [, and the authors use continued fractions as in Stieltjes' original work. In the second paper, where no restriction is required on the support of the measure, the authors study orthogonal Laurent polynomials and closely related Gaussian-type quadrature formulas to obtain the solution, thus following the method of Hamburger. Our approach can be called the Hahn-Banach method, first used by M. Riesz in connection with his solution of the Hamburger moment problem. This method was generalized to higher dimensions by Haviland in the papers cited above and later formulated in terms of adapted spaces by Choquet (1962). Theorem 4.3 is essentially in Wall (1931) although only extensions of s: No ~ [R to {k E 7L I k ~ - 2n} are discussed. That the log-normal distribution is indeterminate for the classical Stieltjes moment problem was noticed by Stieltjes (1894) and Heyde (1963). It seems difficult, but would be very interesting, to determine the structure of perfect semigroups. Theorems 5.4 and 5.5 show two possibilities of building perfect semigroups out of simpler ones. As another possibility let (Sn) be a sequence of semigroups. Then the direct sum 00

S

=

EB Sn =

n=1

{s = (S1' S2" ··)ISnESn' Sn =

o eventually}

is perfect if and only if each Sn is perfect. We sketch the proof and remark 1 This implies in particular that S* first that S* is isomorphic to need not be locally compact for a perfect semigroup S. For ({J E &J(S) and n E N there is a measure iln E E + = 1 S:) such that

flr=

S:.

(flZ

for

Sk E

il E E +

Sk' k

(flr=

1

=

1, ... , n. Itsuffices to prove the existence of a measure

S:) such that il Pn = iln for all n, where

Pn:

00

n

k= 1

k= 1

fl S: ~ fl S:

is the projection, and this is a consequence of a general result about projective systems, cf. Bourbaki (1965-1969, Chap. IX, p. 53). The semigroup of dyadic numbers is perfect, and this may be used to prove that every countable 2-divisible semigroup with the identical involution is perfect, see Berg (1984b). Continuous functions j': Ja, b[ ~ [R such that (x, y)~.l(x + y) is a positive definite kernel on J!a, !b[ x ]!a, !b[ were introduced by Bernstein

Notes and Remarks

225

who called them exponentially convexjilnctions. Both Bernstein and Widder proved Theorem 5.12 in the special case of H being one-dimensional, cf. Akhiezer (1965, p. 210). Akhiezer indicates how the result can be extended to k-dimensional intervals. See also Reeds (1979). Theorem 5.12 is also true if H is an open convex set, cf. Devinatz (1955). His proof uses the spectral theorem for a family of commuting self-adjoint operators. A complete proof of Theorem 5.17 is given in Berg (1984b).

CHAPTER 7

Hoeffding's Inequality and Multivariate Majorization

§1. The Discrete Case Many years ago Hoeffding (1956, Theorem 3) proved the following result: if Xl' ... , X n are independent Bernoulli random variables and if

1/1: {O, 1, ... , n}

--+

IR

is (strictly) concave, then (1) where on the right-hand side it is assumed that IFD(X i = 1) = Pi while on the 1 Pi for all i = 1, ... , n. left-hand side IFD(X i = 1) = P = (lin) Recently, Bickel and van Zwet (1980, Theorem 1.1) found a considerable extension of this result: let 1/1: IR m --+ IR and k E r\J be given and let Xl' , Xn be independent m-dimensional random vectors with distributions ill' , iln and the extra condition that there is a finite subset A ~ IR m (depending on ilb ... , iln) with cardinality k such that ilj(A) = 1 for allj = 1, ... , n. Then the following three conditions are equivalent:

2:7=

(i) the inequality

(2) holds for all n and all such ill' ... , iln; (ii) the inequality (2) holds for n = 2 and all ill' il2 of the type described above; (iii) the k x k matrix (I/I(Xi + Xj))i,j= 1, ... ,k is negative definite for every choice of X b ••• , X k E IR m •

227

§1. The Discrete Case

The interpretation of (2) is analogous to that of (1): on the right-hand side X i has the distribution J.1i while on the left-hand side the random vectors Xl' ... , X n are identically distributed with the average distribution fi = (lin) 1 JJi· Note that (2) reduces to (1) in case of {a, l}-valued random variables. The third condition in Bickel and van Zwet's result makes likely a certain kinship between negative definite functions and functions fulfilling inequalities of type (2) on more general semigroups than Just euclidean spaces. This, in fact, is true and will lead us to interesting characterizations of negative definite functions as well as of completely negative definite functions on arbitrary abelian semigroups. This last-mentioned class of functions has not yet been defined, but appears naturally in this connection, and the lower bounded members of this class turn out to be nothing more than the completely alternating functions. In this chapter we only consider abelian semigroups with the identical involution. First let S be an arbitrary nonempty set. If (Q, .91, P) is a probability space then X: Q ~ S is called an elementary random variable if {X = s} Ed for all S E S and IX(Q) I < 00, i.e. X assumes only finitely many value~. The distribution of an elementary random variable is, of course, a finite convex combination of one-point measures, or, as we have called it earlier, a molecular probability measure. If S is an abelian semigroup and X b . . . , X n are elementary S-valued random variables, then the sum 1 Xi is of this type, too. We shall make repeated use of the following straightforward identity:

L7=

L7=

1.1. Lemma. Let (ajk) be a symmetric real n x n matrix. Thenfor

Cb ... ,

Cn'

d 1, ... , dn E lR we have

~

i..J

j,k= 1 PROOF.

For j

. (C j + d j . Ck + dk _ .d) = a)k 2 2 c) k

~

i..J

j,k= 1

.

a)k

Cj -

2

dj . Ck - dk 2·

(3)

* k the two factors of ajk = akj on the left-hand side sum up to

2 . Cj

+ d j . Ck + dk _ C .dk _ Ck d . = 2 . Cj - d j . Ck - dk 2 2 ) ) 2 2

and the factor of a jj is given by

(

Cj

+ d j)2 2

_ c).d.) = (Cj -2 dj)2 '

thereby proving the equality (3). The next result is now nearly obvious:

D

228

7. Hoeffding's Inequality and Multivariate Majorization

1.2. Proposition. Let S be a nonempty set and let l/!: S x S ~ lR be symmetric. Then jor independent elementary S-valued random variables X and Y the inequality (4) [E;1[l/!(X, Y)J ~ [E1l1,1l2[l/!(X, Y)J holds for all J.11 and J.12 (In (4) we define and (2).)

if and only if l/! is negative definite.

fi = !(J.11 + J.12) and the interpretation is analogous to (1)

Let the two probability distributions J.11 and J.12 be concentrated on {S1' ... , sn} ~ S and denote (Xi = J.11({sil), Pi = J.12({sil). Then by Lemma 1.1

PROOF.

[E;1[l/!(X, Y)] - [E1l1,1l2[l/!(X, Y)]

((Xi

+ Pi (Xj + Pj

i.~ 1 l/J(Si , s) - 2 - - 2 - - (XiP j n

=

(X.-f3.l (X.-f3, l/!(Si' Sj) -l-2- . ~2),

n

.?: l,j=

)

(5)

1

and this is clearly nonpositive if l/! is negative definite. Suppose on the other hand that (4) holds for all J.11' J.12. Let S1' ... , Sn E S and C1 , ••. , Cn E lR with Ci = 0 be given. If not all Ci are zero then 0 < = Ci- =: C, and putting

L ct

L

L

the equality (5) shows n

L

i,j = 1

CiCjl/!(Si, Sj) ~ 0,

D

which had to be shown.

1.3. Definition. Let S be an abelian semigroup. Then l/!: S ~ lR is said to fulfil Hoeffding's inequality oforder n ~ 2 if for every sequence X l ' ... , X n of n independent elementary S-valued random variables the inequality

[E;1[l/!(X 1

+ ... + X n)]

~ [E1l1, ...,lln[l/!(X 1

+ ... + X n)]

holds, where on the right-hand side Xi has the distribution J.1i' 1 ~ i ~ n, while on the left-hand side all the Xi have the same average distribution fi:= (lin) 1 J.1i· The above inequality can also be expressed as

L7=

f l/J dji*" ~ f l/J d(/11 * ·.·* /1,,). Specializing in 1.2 the set S to a semigroup gives 1.4. Corollary. Ij'S is an abelian semigroup and l/!: S ~ lR a givenjunction on S, then l/! fulfils Hoeffding's inequality of order 2 if and only if l/! is negative

definite.

229

§1. The Discrete Case

A negative definite function on a semigroup need not however fulfil Hoeffding's inequalities of a higher order. As an example take S = No with usual addition. The function l/!:= 1{1, 3,5, ...} is negative definite on No because l/!(k) = t[l - (-l)kJ. If X 1, X 2, X 3 are independent Bernoulli random variables with parameters PI = 0, P2 = P3 = i, then p = and

t

[p[l/!(X 1 + X 2

+ X 3 )J = 3p(1

- p)2

+ p3 =!,

but

[Pl' P2, P3[l/!(X 1 + X 2 + X 3)J =

i·

The next result will show that Hoeffding's inequality of order 3 actually implies a sharpened form of negative definiteness.

1.5. Proposition. Let t/J: S -+ [R fulfil Hoeffding's inequality of order n ~ 3. Then for all a E S the translate E (ii)" If l/J is completely negative definite then exp( - tl/J) is completely positive definite, hence Schur-decreasing for t > O. Therefore

PROOF.

t/J = lim ~ (l - e- f"') l-+O

t

is Schur-increasing. "(ii) => (i)" If qJ is completely positive definite then - qJ is completely negative definite and hence Schur-increasing. Every pESt is Schur-decreasing as shown in the proof of Theorem 3.7, and therefore every moment function qJ E .1f(S) with representing measure Il supported by st is Schur-decreasing. This shows that (i) holds for S = No because a completely positive definite function on No is a Stieltjes moment sequence, cf. 6.2.5. Furthermore (i) holds for a perfect semigroup by Lemma

3.8.

[J

3.10. Corollary. Let S be a Hausdorfftopological abelian semigroup containing a dense perject semigroup. Then every continuous completely positive (resp. negative) definite function qJ: S -+ lR is Schur-decreasing (resp. increasing). PROOF. Let two vectors Il = (Ilb . .. , Iln) and v = (VI, ... , vn) of molecular probability measures on S be given such that V ~ Il. There is a finite set A s;; S on which all Ili and hence all Vi are concentrated. Approximating each sEA by a suitable net of points from the given dense perfect subsemigroup P we can find nets Il(i) = (IlY), ... ,Il~i») and v(i) = (vy), ... , V~i») from Mol~(p)n such that v(i) ~ Il(i) and

J qJ

d(Vl

* ... * vn) =

li:n

~ li:n

J J qJ

d(vY)

*

* v~i)

qJ

d(J.LY)

*

* J.L~» =

J qJ

d(J.Ll

* ···* J.Ln),

where we have used that qJ IP is completely positive definite and hence Schur-decreasing. The case of qJ being completely negative definite is treated similarly. D This corollary applies, in particular, to the important semigroups (IR~, +) and (lR\ +). Since they are 2-divisible, we conclude that continuous negative definite functions are the same as continuous Schur-increasing functions on these semigroups.

246

7. Hoeffding's Inequality and Multivariate Majorization

In Proposition 1.11 we saw that a nonnegative quadratic form q on an abelian group G fulfils the reverse Hoeffding inequality of each order. As we shall see now q is even Schur-decreasing. It should be noticed that G is considered as a semigroup with the identical involution, so there is no contradiction in the fact that q is negative definite in the group sense and -q is completely negative definite in the semigroup sense.

3.11. Proposition. A nonnegative quadratic form q on an abelian group G is Sc hur-decreasing.

= 7L k for some k

PROOF. We consider first the case where G

~ 1. In the proof

of Theorem 4.3.9 it was shown that B(s, t)

:=

q(s)

+ q(t)

- q(s - t),

is biadditive and, furthermore, that B is a positive definite kernel. Denoting e1 = (1, 0, ... ,0), ... ,ek = (0, ... ,0, 1) we see, in particular, that the matrix (B(ei' ej))i,j= 1, ... ,k is positive definite. Noting q(O) = this implies

°

k

q(s)

=

=

tB(s, s)

tL

i,j= 1

sisjB(ej, ej)

and q is therefore the restriction of a continuous nonnegative quadratic form on IRk. The result follows now from 3.10 and 1.11 since IRk contains the dense perfect semigroup Qk. If G is arbitrary, let J.11' .•• , J.1n be all concentrated on the finite subset A = {S1"." Sk} S;; S implying vi(A) = 1 for all i = 1, ... , n if v -< J.1. We define a canonical group homomorphism h: 7l. k ~ G by h(z 1, ... ,Zk) := k L~= 1 ZjSj' Then q 0 h is a nonnegative quadratic form on 7L • Let ii and v k on 7L be defined by k

iii:=

k

L J.1iC {Sj} )eej'

Vi:=

j= 1

j= 1

Then ji~

= J.1i'

v~

=

Vi

for i

L viC {Sj} )eej'

= 1, ... , n and

f q d(Vl * ·..

*V n )

= =

=

~

V -< ji, implying

f q d(v~ * ·.. * v~)

f

q d[ (v 1 * ...

fq

* Vn)h]

h d(Vl

* .·.* vn)

q 0 h d(ill

* ... * iln)

0

f f

= q d(J.ll * .. ·* J.ln), where we have used the result of Exercise 2.3.12.

o

§3. Completely Negative Definite Functions and Schur-Monotonicity

247

3.12. As in the case of Hoeffding's inequality we can extend the results proved so far in this section to nondiscrete distributions, assuming, of course, continuity of the positive and negative definite functions (kernels) to be integrated, and assuming furthermore that the probabilities involved are Radon measures. To prove these extensions one uses the same ideas as in §2; cf., in particular, the proofs of Theorems 2.1 and 2.5. We will now show the existence of completely negative definite functions on N6 which are not Schur-increasing. By Theorem 3.9 this is equivalent with the existence of completely positive definite functions on N6 which are not Schur-decreasing. We prove even more: 3.13. Proposition. There exist completely positive definite functions on N6 which are not Schur-decreasing oj'order 3. PROOF.

Let B denote the set of polynomials

p(x, y) =

,

~

. k

CjkX)y

L

in two variables where cjk ~ 0 and p(l, 1) = Cjk = 1. In accordance with Definition 3.2 we consider the ordering -< on B 3 and define the set 13 = {qlq2q3 - PIP2P31p = (Pb P2' P3), q = (ql' q2' q3) E B 3, q -< p}. For a function qJ: N6

-+ ~

and a polynomial P E O. (iv) Put ~al/J(h'):= l/J(a + h') - t/J(h')jor h' E H' and a E S. Then ifl/J we get ~at/J E gpq(H)jor all a E S.

E

%q(H)

PROOF. (ii) is a trivial property. (iii) follows from Theorem 3.2.2. (iii) together with Lemma 1.1 implies (i). Concerning (iv) we observe that if t/J E -f!JJ'l(H) then ~at/J E f!JJ'l(H) by Lemma 1.1. Using the closedness of f!JJ'l(H) and the fact that

IjJ = lim ~ [l - exp( - tljJ)], 1-0

we get

~a

t

l/J E f?j>q( H) for every l/J E JVq( H).

D

Let us consider for a moment the semigroup S with neutral element. If

l/J E %(S) is bounded below and ljJ(s)

= c + a(s) +

r

(l - p(s» dJi(p)

JS\{l}

is the uniquely determined representation given by Theorem 4.3.20, where CE~, rx:S-+ ~+ is additive and J.1EM+(S\{l}), and if ~at/J(s) = t/J(s + a) - l/J(s) = fja(s), then (1)

and (1 - p(a))J.1

=

(ja

on S\{l},

implying J.1({1{0}}) = (ja({l{O}}) for all a E S\{O} = H. Motivated by this result and taking into account the last lemma we define %fi(H) := {t/J

E

%q(H) I~at/J E gp'6(H) for aU a E H}.

258

80 Positive and Negative Definite Functions on Abelian Semigroups Without Zero

It is easily seen that JVfi(H) is an extreme subcone of J¥°q(H), and from Lemma 1.2 it follows immediately that the two cones coincide in case H is without maximal elements.

1.9. Theorem. Let t/J E JV'6(H) be given. Then there exist an additive function (1:S -+ ~+ and a Radon measure J1 on the locally compact space H\{I} such that for h', k' E H' ljJ(h') - ljJ(k')

= 1_

(p(k') - p(h'» dJ1(p)

+ rx(h')

- rx(k').

JH\{l}

The pair «(1, J1) is uniquely determined and given in the following way:

+ ns)

t/J(2h

(1() s = 11m 0

n

n-cc

independent oj' h E H, (!)a = {p E Hlp(a) < I},

on the open set where (Ja is the Radon measure on H satisfying AaljJ(h')

h'EH'.

= LiP(h') d(Ja(P),

PROOF. From ~al/JEg;q(H) we get, using Lemma 1.8, that and hence ~b( -~at/J) E 9;q(H). We have

-~at/JEJVq(H)

+ b) + ~at/J(h') + b + h') + t/J(h' + b) + t/J(h' + a)

[~b( -~at/J)](h') = -~at/J(h'

= -t/J(a

= L/(h') d(Ja(P) - L/(h'

= L,P(h')[l - p(b)] d(Ja(P)

+

- t/J(h')

b) d(Ja(P)

= L,P(h')[l - p(a)] d(Jb(P)

for all h' E H', and the unicity statement in Theorem 1.4 tells us that

[1 - p(b)](Ja = [1 - p(a)](Jb

on

Taking into account that the open sets (!) a' a E H, cover again Theorem 2.1.18 we conclude that

J1 := [1 - p(a)] - 1 (J a

on

H. H\ {I}, and invoking

(!) a

is a well-defined Radon measure on H\{l} satisfying [1 - p(a)]J1 = (Ja on H\{l}. Hence AaljJ(2h)

=

Iv JH\{l}

for all h E H and a E S.

p 2 (h)[l - p(a)] dJ1(p)

+ (Ja({l})

259

§l. Quasibounded Positive and Negative Definite Functions

Now we observe that ~st/J(2h) = t/J(2h + s) - t/J(2h) =: t/Jh(S) is nonnegative and negative definite as a function of s; hence t/J h has the representation ljIh(S)

= ah(s) + fA

[1 - p(s)] dJ1.h(P)

JS\{l}

with an additive function (Xh: S

-+

+ s)

f,

ljI(2h

- ljI(2h)

=

~+ and f.1h

E

M +(S\{l}). We get

p2(h)[1 - p(s)] dJ1.(p)

+ o-s({I})

JH\{l}

f

= ah(s) +

[l - p(s)] dJ1.h(P)

JS\{l}

for all hE H, S E S. From relation (1) we learn that (Xh(S) = !s,h({l}) where is,h = ~sljJh' Put h' = 2h + a, then ~st/J(h')

= t/J(2h + a + s) - t/J(2h + a) =

~st/Jh(a)

= L-/(h') do-s(p) = f/(a)p2(h) do-s(p), and therefore !s, h = p2(h)O's implying (Xh(S) = !s.h({I}) = O's({I}), independent of h and additive in S. Put (X(s) := O's( {I}), then indeed

SE

ljI(h'

+ s) - ljI(h')

=

=

t!sljl(h') = Lp(h') do-s(p)

f,

p(h') do-.(p)

+ o-s({I})

JH\{l}

=

f,

p(h')[1 - p(s)] dJ1.(p)

+ a(s)

JH\{1}

for all h' E H' and S E S. Computing in this way t/J(h' ljJ(h' + k') - t/J(k') for h', k' E H' gives us, finally, ljI(h') - ljI(k')

=

f,

[p(k') - p(h')] dJ1.(p)

JH\{l}

+ k')

+ a(h')

- t/J(h') and

- a(k')

as asserted. From Theorem 4.3.20 we know that a(s)

=

ah(s)

= lim! ljIh(ns) = lim! ljI(2h + ns) n-+oo

n

n-+oo

n

independent of h E H. Hence (X is uniquely determined and so then is f.1, because any representing measure must coincide on (!Ja with [1 - p(a)]-lO'a D by immediate calculation, using the unicity statement in Theorem 1.4.

260

8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero

The integral representations of 1.4 and 1.9 lead naturally to the question about a converse statement, to be answered now. 1.10. Proposition. Put Xh'(P):= p(h')jor h' E H' and P E

H.

(i) If J.1 is a Radon measure on fl such that Xh' E !£1(J.1) for each h' E H', then qJ: H' -+ IR defined by qJ(h') := Xh' dJ.1 belongs to f!JJ1J(H). (ii) Ij' J.1 is a Radon measure on H\ {I} such that Xh' - Xk' E fLJI(J.1) jor all h', k' E H', and if'a: S -+ IR+ is additive, then t/J: H' -+ IR defined by

Iii

r.

t/J(h'):=

+ rx(h'),

(Xk' - Xh') dJi

JH\{l}

k' E H' arbitrary but fixed, belongs to %'6(H). PROOF. (i) Of course

suplqJ(h' seS

qJ is positive definite and for each h' E H'

+ s)1 ~

+ s)1 dJ.1(p) ~

sup fIP(h' seS

fIP(h')1 dJ.1(p)


f

[p(k') - p(2h 1 )] dJi(p)

-00

2

2

+t

f

[p(k') - p(2h 2 )] dJi(p)

261

§l. Quasibounded Positive and Negative Definite Functions

for all

SE

S; hence

l/J is quasibounded below. Finally, for a E Sand h' E H'

~aljJ(h') = r~

[1 - p(a)]p(h') dJi(p)

+ o:(a)

JH\{l}

=

L/(h') daa(p),

where aa(P) = [1 - p(a)]Jl on H\{l} and aa({l}) = Cl(a), implying ~al/J E f7Jb(H) by (i), hence indeed l/J E %b(H) as asserted. 0 Looking once more at the proof of 1.9, we see that only the property Ia E H} £; &'b(H) has been used. Together with the above proposition this means, that any function l/J: H' --+ ~ such that ~a l/J E &'b(H) for all a E H belongs already to %b(H), a partial converse to Lemma 1.8(iv). {~al/J

1.11. Example. Let again H be the additive semigroup ]0, 00[. In 1.5 we saw that fI can be identified with [0, 00 [, where the point corresponds to the semicharacter 1, hence H\{l} ~ ]0,00[. We have JVb(H) = %q(H) because H is 2-divisible. By Theorem 1.9 any negative definite function l/J: ]0, 00 [ --+ ~, quasibounded below, has the representation

°

ljJ(s) - ljJ(t)

=

r J]O,

(e- A'

-

e- AS) dJi()')

+ o:(s -

t)

oo[

for all s, t E ]0, 00 [, where Cl E IR + and Jl E M + (]O, 00 E). From this and Proposition 1.10 it follows that the functions in JVq(]O, 00 [) can be characterized as

where

ljJ(s)

=

IR, Cl ~

°

dJi()')
0,

(2)

and Jl E M +(]O, oo[) is such that

From (2) we see that

f"

and

e- SA dJi()')
0.

l/J is Coo and that ljJ'(s)

=

0:

+

{Xl

).e- AS

dJi()')

is completely monotone. Conversely, if t/J: ]0, 00 [ -+ IR is Coo and t/J' is completely monotone, we have by Corollary 4.6.14 that l/J' = !fa for some a E M + ([0, 00 [) and therefore ljJ'(s)

=

0:

+

{Xlk -

AS

dJi().),

with Cl = a( {O}) and Jl = A-I (a 1]0, 00 C), hence longs to %q(]O, 00 E).

l/J has the form (2) and be-

262

8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero

An important example is given by the log-function whose integral representation reads log s - log

=

t

[00

J

o

(e- A1

e- AS )

-

dA

T'

s, t > 0,

or, taking t = 1, s>

o.

1.12. Example. For the additive semigroup N we found in 1.6 that N ~ [-1, 1]\{0}, hence N\{I} ~ [-1, 1]\{0, I}. A measure J.1 on N\{I} for which Proposition 1.10(ii) can be applied, is, for example, given by dt

The corresponding negative definite function tjJ: N' the values (choosing k' = 2) tjJ(2)

= 0,

tjJ(3)

t/J(2n

= tjJ(4) = 1,

+ 1) =

t/J(2n

tjJ(5)

=

{2, 3,4, ... } -+

~

has

= tjJ(6) = 1 + t, ... , 1

+ 2) = 1 + t + t + ... + -. n

The log-function restricted to N' is of course also negative definite there. It has the integral representation log n - log m =

f

(t m

-

tn)

]O,l[

From the description of q>q(] 0, 00 [) and JV'l(]O, 1.5 and 1.11 we have the following result:

-t

dt I . ogt

00 [)

1.13. Theorem. Ij' qJ E &>'l(]0, oo[) then tjJ: ]0, oo[

given in the Examples

-+ ~

defined by tjJ(s):=

J1 qJ(t) dt belongs to JVq(]O, oo[). Conversely, the derivative oj' any jiinction in JVq(]O,

00 [)

belongs to g>q(]O,

00

E).

1.14. Exercise. Let H be N k with componentwise addition. Show that fI can be identified with ([ -1, O[ u ]0, 1])k. Show that both n = (n t , ••• , nk) ~ (nt+···+nk)-l and n~(-l)nl+···+nl'(nl+···+nk)-t are positive definite (and quasibounded) and try to find their integral representations. 1.15. Exercise. Let H be the real line with maximum operation. Then g>q(H) is the set of all nonnegative decreasing functions, and JVq(H) consists of all increasing functions on ~. The same is true if we replace the line by an arbitrary nonempty subset of ~.

~2.

263

Completely Monotone and Completely Alternating Functions

1.16. Exercise. If H is the additive semigroup JO, oo[n, then FJ == [R~ (by natural identification). Show that all first partial derivatives of a function in %q(]O, 00 en) belong to gPq(]O, 00 en). 1.17. Exercise. Show that IIxll- I is not positive definite on ]0, oo[n for n > 1, 11·11

denoting the euclidean norm.

1.18. Exercise. Let l/J be a strictly positive, negative definite function on the abelian semigroup H. Then if j'E gPq(]O, oo[) we have f l/J E gPq(H), and if g E%q(]O, oo[), then g ffE %q(H). 0

0

1.19. Exercise . Use 1.18 to show that x 1---+ log(2:7 = I xJ is negative definite 00 [" and that x 1---+ (2:7=1 xJ - t is positive definite for all t > 0.

on JO,

1.20. Exercise. Let K be a compact Hausdorff space and let H be the set of all strictly positive continuous functions on K with pointwise addition. Show that H can be identified with M +(K), the set of Radon measures on K. 1.21. Exercise. Let l/J E %6(H) have the representation given in Theorem 1.9. Then l/J can be extended to a negative definite function on S, bounded below, if and only if SH\{l} [1 - p(h)] dp,(p) < 00 for all hE H.

°

1.22. Exercise. Let S be a perfect semigroup with zero element and identical involution, and assume that H:== S\{O} fulfils H == H + H. Then H is a perfect semigroup without zero in the sense that for every qJ E f2P(H) there is a unique Radon measure J1 on S* \ r 1{a}} such that qJ(h)

=

fp(h)d~(p),

hEH.

§2. Completely Monotone and Completely Alternating Functions Again let (H, +) denote an arbitrary abelian semigroup and put, as in the preceding section, S :== H u {a}, H :== S\ {I fO}}. In the definition of completely monotone (resp. alternating) functions on H no change is necessary compared with the definition for functions on semigroups with neutral element (cf. 4.6.1): we again require (resp.

~

0)

for all n E N, (hI' . .. ,hn) E Hn and all hE H. If one of the h/s is 0, this expression is still well defined and equal to zero, so that the condition is not changed by allowing (hI' ... , hn ) E sn. Of course a completely monotone function is also required to be nonnegative.

264

8" Positive and Negative Definite Functions on Abelian Semigroups Without Zero

We denote by ---'I(H) (resp. d(H)) the set of all completely monotone (resp. alternating) functions on H. Note that in contrast with the positive and negative definite functions which in a natural way are only defined on the subsemigroup H' of nonmaximal elements of H, the completely monotone (resp. alternating) functions are defined on all of H. For f: H -+ [R and h E H let fh: S -+ [R be defined by fh(S) := f(h + s), and for fixed s E S define ~sf: H -+ [R by ~sf(h) := f(h + s) - f(h). Let us first state some simple properties of ---'I(H) and d(H). 2.1. Lemma. (i) Both ---'I(H) and d(H) are closed convex cones in [RH. (ii) - ---'I(H) ~ d(H). (iii) A function qJ: H -+ [R belongs to ---'I(H) (resp. d(H)) if and only

---'I(S) (resp. qJh E d(S))for all hE H. (iv) A function t/J: H -+ ~ belongs to d(H) if and only function ~a t/J belongs to ---'I(H). (v) qJb qJ2 E ---'I(H) implies qJl . qJ2 E ---'I(H). (vi) l/J E d(H) if"and only if e-tl/J E ---'I(H)jor all t > O. PROOF.

if qJh E

if jor every a E S the

(i) and (ii) need no proof. (iii) follows from the identity

Val··· VanqJh(S) = Val··· VanqJ(h valid for all hE H, S E S, n E N, (a 1 , We get (iv) from

••• ,

+ s)

an) E sn.

Val··· Van(~al/J) = - Val··· VanVal/J, which holds for all a E S, n E N and (at, ... , an) E sn. Using (iii) and the fact that ---'I(S) is closed under pointwise multiplication we get property (v) observing also that (qJl . qJ2)h = (qJt)h . (qJ2)h. In the same D way (vi) follows from Proposition 4.6.10. Let qJ be a completely monotone function on H. Then the above lemma tells us that for each h E H the translate qJh belongs to ---'I(S) implying by Theorem 4.6.5 the representation benhavns Universitet. JONES, W. B., W. J. THRON and H. WAADELAND (1980). A strong Stieltjes moment problem. Trans. Amer. Math. Soc. 261, 503-528. JONES, W. B., O. NJASTAD and W. J. THRON (1984). Orthogonal Laurent polynomials and the strong Hamburger moment problem. J. Math. Anal. Appl. 98, 528-554. KAHANE, J.-P. (1979). Sur les fonctions de type positif et de type negatif. Seminar on Harmonic Analysis 1978-79, pp. 21-37. Publ. Math. Orsay 79, No.7, Univ. Paris XI,Orsay. KAHANE, J.-P. (1981). Helices et quasi-helices. In Mathematical Analysis and Applications (Ed. by L. Nachbin). Essays dedicated to Laurent Schwartz on the occasion of this 65th birthday. Part B. Adv. Math. Suppl. Studies 7B. New York-London: Academic Press.

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SCHOENBERG, I. J. (1938b). Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522-536. SCHOENBERG, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9, 96-108. SCHUR, I. (1911). Bemerkungen zur Theorie der beschrankten Bilinearformen mit unendlich vielen Veranderlichen. J. Reine Angew. Math. 140, 1-29. SCHWARTZ, L. (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. London: Oxford University Press. SHOHAT, J. A. and J. D. TAMARKIN (1943). The Problem of Moments. Math. Surveys 1. Providence: American Mathematical Society. SIBONY, D. (1967-1968). Cones de fonctions et potentiels. Cours de 3eme cycle it. la Faculte des Sciences de Paris. STEWART, J. (1976). Positive definite functions and generalizations, a historical survey. Rocky Mountain J. Math. 6, 409-434. STIELTJES, T. J. (1894). Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1-122, and 9, 1-47. (In Oeuvres Completes, Vol. II, pp. 402-567. Groningen: Noordhoff, 1918.) STOCHEL, J. (1983). The Bochner type theorem for *-definite kernels on abelian *-semigroups without neutral element. In Dilation Theory, Toeplitz Operators, and other Topics, pp. 345-362. (Ed. by G. Arsene). Basel-Boston-Stuttgart: Birkhauser Verlag. SVECOV, K. I. (1939). On Hamburger's moment problem with the supplementary requirement that masses are absent on a given interval. Commun. Soc. Math. Kharkov 16, 121-128 (in Russian). SZAFRANIEC, F. H. (1977). Dilations on involution semigroups. Proc. Amer. Math. Soc. 66,30-32. SZANKOWSKI, A. (1974). On Dvoretzky's theorem on almost spherical sections of convex bodies. Israel J. Math. 17, 325-338. SZ.-NAGY, B. (1960). Extension of Linear Transformations in Hilbert Space which extend beyond this Space. New York: Ungar. TALAGRAND, M. (1976). Quelques examples de representation integrale: Valuations, fonctions alternees d'ordre infini. Bull. Sci. Math. 100 (2), 321-329. TONEV, T. W. (1979). Positive-definite functions on discrete commutative semigroups. Semigroup Forum 17,175-183. TOPS~E, F. (1970). Topology and measure. Lecture Notes in Mathematics 133, BerlinHeidelberg-New York: Springer-Verlag. TOPS"(S) IqJ(O) = I} &:.(B) %(S) negative definite functions %'(S) lower bounded negative definite functions %b(S) bounded negative definite functions A(S) completely monotone functions A 1(S) = {qJ E A(S) IqJ(O) = I} d(S) completely alternating functions Yr(S) moment functions E+(S*) = {,u E M +(S*)IJ IXsl d,u < 00 for all s E S} E+(S*, qJ) = {,u E E+(S*)IJ XS d,u = qJ(s) for all S E S} E(S*) = {,ul - ,u2 + i(,u3 - ,u4)I,ul,···,,u4 E E+(S*)} 2 set of Levy measures

129 91, 123 110, 129, 130 103 130 130 92 96

180 181 88 88 94 94 94

96 96

153 89 99

105 130 130 179 179

180 203 105

6. Notations Related to an Abelian Semigroup (H, +) which (Possibly) has no Zero (S:= H u {O}) H' = H + H

fJ = S\ {l{o}} 9(H)

the nonmaximal elements of H

&"l(H)

positive definite functions on H quasibounded positive definite functions

96(H) %(H) %q(H)

negative definite functions on H negative definite functions, quasibounded below

%~(H)

A(H) Ao(H) d(H) do(H)

completely monotone functions on H completely alternating functions on H

252 255 253 253 254 253 257 257 264 264 264

265

284

List of Symbols

7. Miscellaneous Symbols 3

PA

seminorm determined by A supremum norm E* algebraic dual space E' topological dual space alg span(r), alg span + ( r) r-positive linear functionals

II . 1100

A:

/P

= {x E IR

'pP(Jl) LP(Jl)

=

I I IJI Ix(n)IP < oo}, 0 < p