Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Universitext

Editorial Board (North America):

S. Axler F.W. Gehring P.R. Halmos

U niversitext

Editors (North America): S. Axler, F.W. Gehring, and P.R. Halmos

Aksoy/Khamsi: Nonstandard Methods in Fixed Point Theory

Aupetit: A Primer on Spectral Theory

Booss/Bleecker: Topology and Analysis

Borkar: Probability Theory; An Advanced Course

Carleson/Gamelin: Complex Dynamics

Cecil: Lie Sphere Geometry: With Applications to Submanifolds

Chae: Lebesgue Integration (2nd ed.)

Charlap: Bieberbach Groups and Flat Manifolds

Chern: Complex Manifolds Without Potential Theory

Cohn: A Classical Invitation to Algebraic Numbers and Class Fields

Curtis: Abstract Linear Algebra

Curtis: Matrix Groups

DiBenedetto: Degenerate Parabolic Equations

Dimca: Singularities and Topology of Hypersurfaces

Edwards: A Formal Background to Mathematics I alb

Edwards: A Formal Background to Mathematics II alb Foulds: Graph Theory Applications

Fuhrmann: A Polynomial Approach to Linear Algebra

Gardiner: A First Course in Group Theory

Girding/Tambour: Algebra for Computer Science

Goldblatt: Orthogonality and Spacetime Geometry

Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups

Holmgren: A First Course in Discrete Dynamical Systems

Howe/Tan: Non-Abelian Harmonic Analysis: Applications of 5L(2. R)

Howes: Modern Analysis and Topology

Humi/Miller: Second Course in Ordinary Differential Equations

Hurwitz/Kritikos: Lectures on Number Theory

Jennings: Modern Geometry with Applications

Jones/Morris/Pearson: Abstract Algebra and Famous Impossibilities

Kannan/Krueger: Advanced Real Analysis

Kelly /Matthews: The Non-Euclidean Hyperbolic Plane

Kostrikin: Introduction to Algebra

Luecking/Rubel: Complex Analysis: A Functional Analysis Approach

MacLane/Moerdijk: Sheaves in Geometry and Logic

Marcus: Number Fields

McCarthy: Introduction to Arithmetical Functions

Meyer: Essential Mathematics for Applied Fields

Mines/Richman/Ruitenburg: A Course in Constructive Algebra

Moise: Introductory Problems Course in Analysis and Topology

Morris: Introduction to Game Theory

Porter/Woods: Extensions and Absolutes of Hausdorff Spaces

Ramsay /Richtmyer: Introduction to Hyperbolic Geometry

Reisel: Elementary Theory of Metric Spaces

Rickart: Natural Function Algebras

Rotman: Galois Theory

Rubel/Colliander: Entire and Meromorphic Functions

Sagan: Space-Filling Curves

Samelson: Notes on Lie Algebr'ts

(continued following index)

Michel Simonnet

Measures and Probabilities

Foreword by Charles-Michel Marie

Springer

Michel Simonnet Department of Mathematics University of Dakar Senegal

Editorial Board (North America):

S. Axler Department of

Mathematics Michigan State University East Lansing, MI 48824 USA

F. W. Gehring Department of

Mathematics University of Michigan Ann Arbor, MI 48109 USA

AMS Subject Classification (1991): 28-02, 28c05, 60Fxx

Library of Congress Cataloging-in-Publication Data Simonnet, Michel

Measures and probablilities / Michel Simonnet. p. cm - (Universitext)

Includes bibliographical references and indexes.

TSBN-13:978-0-387-94644-3 e-TSBN-13:978-1-4612-40 12-9 DOl: 10.1007/978-1-4612-4012-9

1. Measure theory. 2. Probabilities. 1. Title. QA312.S54 1996 515'.783-dc20 95-49240

Printed on acid-free paper.

@ 1996 Springer-Verlag New York, Inc.

P.R. Halmos Department of

Mathematics Santa Clara University Santa Clara, CA 95053 USA

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York. Inc .. 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks. etc .. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Robert Wexler; manufacturing supervised by Joe Quatela. Photocomposed copy prepared using the author's I;\'IEX file.

987654321

SPIN 10523961

Foreword

Integration theory holds a prime position, whether in pure mathematics or in various fields of applied mathematics. It plays a central role in analysis; it is the basis of probability theory and provides an indispensable tool in mathe­matical physics, in particular in quantum mechanics and statistical mechanics. Therefore, many textbooks devoted to integration theory are already avail­able. The present book by Michel Simonnet differs from the previous texts in many respects, and, for that reason, it is to be particularly recommended.

When dealing with integration theory, some authors choose, as a starting point, the notion of a measure on a family of subsets of a set; this approach is especially well suited to applications in probability theory. Other authors prefer to start with the notion of Radon measure (a continuous linear func­tional on the space of continuous functions with compact support on a locally compact space) because it plays an important role in analysis and prepares for the study of distribution theory. Starting off with the notion of Daniell measure, Mr. Simonnet provides a unified treatment of these two approaches. Both integration theories (the one with respect to a set-type measure or the other with respect to a Radon measure) are deduced as particular cases of integration with respect to a Daniell measure. Incidentally, theorems on ex­tension of a measure initially defined on a Boolean semiring of subsets of a set are obtained as direct consequences of properties of the integral with respect to a Daniell measure. Thus the author has been able to offer developments of the theory that are of interest to mathematical analysis, and others that are relevant to probability theory. Furthermore, this exposition has the advantage that it leads very quickly to some important results of constant use, such as convergence theorems.

VI Foreword

On the other hand, the framework chosen here is fairly more general than the one usually adopted: measures with respect to which one integrates can be real of either sign or even complex: from the outset, the integral is defined for functions taking their values in a Banach space, as well as for real- (or extended real-) valued functions.

Finally, the treatment of the subject given in this book is much more com­prehensive than the treatment found in most other texts of the same level. The important aspects of integration theory (LP spaces, different types of conver­gence, measure decomposition, Radon-Nikodym derivatives, image measures and product measures) are all treated with great care. The book contains an excellent introduction to probability theory, covering the strong law of large numbers, the central limit theorem, and conditional probabilities. It also pro­vides some applications of integration in analysis, such as the definition of Haar measure on locally compact groups and the convolution of measures.

The book is quite easily accessible thanks to the inclusion by the author of most of the necessary background in topology and analysis. Clarity of the exposition and numerous exercises at the ends of the chapters allow it to be used as a textbook. It will also prove very useful to teachers and will provide a good source of reference for many mathematicians.

CHARLEs--MrCHEL MARLE

Universite Paris VI Pierre et Marie Curie Mathematiques

Preface

This book is intended to be an introductory, yet sophisticated, treatment of measure theory. It should provide an in-depth reference for the practicing mathematician. Having said this, I hope that advanced students as well as instructors will find it useful.

The first part of this book (Chapters 1 to 8) should prove useful to both analysts and probabilists. One may treat the second and third parts (Chap­ters 9 to 14 and 15 to 18) as an introduction to the theory of probability, or use the fourth part as an introduction to analysis.

I generally proceed from first principles and my treatment is, for the most part, self-contained. Other than familiarity with general topology, some func­tional analysis, and a certain degree of mathematical sophistication, little is required for profitable reading of this text. The topological background needed is covered, for example, in Modern General Topology by J. Nagata. Functional Analysis by W. Rudin will provide the reader with most of the results on topological vector spaces used throughout this text. All other results may be found in A Course in Functional Analysis by J. B. Conway. At the end of each chapter, exercises, which are designed to present some additional material and examples, are provided.

The treatment of integration at the elementary level usually starts with either positive or signed measures on a-algebras or with complex Radon mea­sures on locally compact Hausdorff spaces. In this book, we first consider complex Daniell measures on a space 1t(D., C) of elementary functions. To define the upper integral, r f dV j.L, of any positive function f with respect to the variation V j.L of a complex Daniell measure j.L, a class .:J of positive functions, that contains the upper envelopes of increasing sequences of posi-

viii Preface

tive elementary functions, is introduced. Given a Banach space F, a function f from n into F is J.L-integrable if and only if, for each c > 0, there exists a decomposable function g such that r If - gl dV J.L is less than c. Then are defined the J.L-measurable functions taking their values in a metrizable space. This procedure works, in particular, when H(n, C) is the set of simple func­tions on a semiring S. It also works when n is a locally compact space-not necessarily a-compact-and when H(n, C) is the set of continuous functions on n with compact support. Two fairly general and easy to handle integration theories are thus obtained. The first one (Chapters 9 to 14) is especially suited to the needs of probabilists, the second one (Chapters 19 to 24) to those of analysts. Incidentally, note that many major results on Radon measures fol­low immediately from their counterpart results on abstract measures defined on the semi ring of differences of compact sets.

This present volume is strongly influenced by, and owes much to, Probabil­ity and Measures by P. Billingsley (Wiley, 1985), Integration by N. Bourbaki (Hermann, 1965), Treatise on Analysis (volume 2) by J. Dieudonne (Aca­demic Press, 1976), Measure Theory by P. Halmos (Springer-Verlag, 1974), Mesures et probabilites by C. M. Marle (Hermann, 1974), and Real and Com­plex Analysis by W. Rudin (McGraw-Hill, 1966). The fourth part follows closely N. Bourbaki's treatise.

The courses given by Thi'si Ho Van in measure theory at the University of Liege and the University of Abidjan have provided the motivation and impetus for this exposition of the theory. Note that this list is by no means exhaustive.

I wish to express my gratitude to R. Descombes, J. Dieudonne, C.M. Marle from Paris University, W. Rudin of The University of Wisconsin, and to K. Vo­Khac from Orleans University for their kind remarks and encouragement. I am especially indebted to Professor J. Dieudonne for his longtime support and good counsel and to Professor C. M. Marle for his invaluable suggestions.

Finally I would like to thank P. Toppo and G. Vinel for helping with the corrections, R. Eastaway-Gine and M. Gamar for typesetting the manuscript in g\1EX, and Springer-Verlag for accepting the manuscript. The chapter in­troductions are due to G. Vinel.

Contents

Foreword

Preface

I Integration Relative to Daniell Measures

1 Riesz Spaces 1.1 Ordered Groups ...... . 1.2 Riesz Spaces ........ . 1.3 Order Dual of a Riesz Space 1.4 Daniell Measures ...... .

2 Measures on Semirings 2.1 Semirings, Rings, and u-Rings 2.2 Measures on Semirings ..... 2.3 Lebesgue Measure on an Interval

3 Integrable and Measurable Functions 3.1 Upper Integral of a Positive Function 3.2 Convergence Theorems 3.3 Integrable Sets . . . . 3.4 u-Measurable Spaces . . 3.5 Measurable Mappings . 3.6 Essentially Integrable Mappings

v

vii

1

3 4 9

12 16

26 26 30 38

40 41 49 56 60 62 66

x Contents

3.7 Upper and Lower Integrals 3.8 Atoms ....... 3.9 Prolongations of f.L • .

4 Lebesgue Measure on R 4.1 Base-b Expansions of a Real Number. 4.2 The Cantor Singular Function .. 4.3 Example of a Nonmeasurable Set .

5 LP Spaces 5.1 Definition of LP Spaces 5.2 Convergence Theorems 5.3 Convergence in Measure . 5.4 Uniformly Integrable Sets

6 Integrable Functions for Measures on Semirings 6.1 Measurability ........... 6.2 Complements on the LP Spaces . 6.3 Measures Defined by Masses 6.4 Prolongations of a Measure

7 Radon Measures 7.1 Locally Compact Spaces . 7.2 Radon Measures ..... 7.3 Regularity of Radon Measures 7.4 Lusin Measurable Mappings. 7.5 Atomic Radon Measures . 7.6 The Riemann Integral 7.7 Weak Convergence 7.8 Tight Sequences

8 Regularity 8.1 Regular Measures

70 78 81

86 86 88 90

93 94 99

104 108

121 121 126 127 128

133 134 137 140 143 145 147 153 157

166 . 166

II Operations on Measures Defined on Semirings 173

9 Induced Measures and Product Measures 175 9.1 Measure Induced on a Measurable Set 175 9.2 Fubini's Theorem. . . . . 178 9.3 Lebesgue Measure on Rk . 187

10 Radon-Nikodym Derivatives 193 10.1 Sums of Measures . . . . . 194 10.2 Locally Integrable Functions . 195 10.3 The Radon-Nikodym Theorem 200

10.4 Combination of Operations on Measures . . . 10.5 Duality of LP Spaces. . . . . . . . . . . ... 10.6 The Yosida-Hewitt Decomposition Theorem

11 Images of Measures 11.1 {L-Suited Pairs ....... . 11.2 Infinite Product of Measures 11.3 Change of Variable ..... . 11.4 Elements of Ergodic Theory.

12 Change of Variables 12.1 Differentiation in Rk ....... . 12.2 The Modulus of an Automorphism 12.3 Change of Variables 12.4 Polar Coordinates

13 Stieltjes Integral 13.1 Functions of Bounded Variation 13.2 Stieltjes Measures .. . . . . . . 13.3 Line Integrals . . . . . . . . . . . 13.4 The Lebesgue Decomposition of a Function 13.5 Upper and Lower Derivatives ....... .

14 The Fourier Transform in R k

14.1 Measures in Rk ..... 14.2 Distribution Functions. 14.3 Covariance Matrix ... 14.4 The Fourier Transform. 14.5 Normal Laws in Rn ..

Contents Xl

205 207

· 210

224 225 228 233 236

242 242 248 250 252

265 266

· 268 · 274

277 · 282

286 287 293 294 296

.303

III Convergence of Random Variables; Conditional Expectation 309

15 The Strong Law of Large Numbers 311 15.1 Convergence in Probability . . . . ........ 312 15.2 Independence of Random Variables. . . . . . . . · 313 15.3 An Example of Independent Random Variables. · 316 15.4 The One-Sided Shift Transformation . 318 15.5 Borel's Normal Number Theorem. · 320

16 The Central Limit Theorem 326 16.1 Convergence in Law ... 326 16.2 The Lindeberg Theorem . 330 16.3 The Central Limit Theorem . 334

Xll Contents

17 Order Statistics 17.1 Definition of the Order Statistics 17.2 Convergence of the Empirical Median

344 344

. 347

18 Conditional Probability 352 18.1 Conditional Expectation. . . . . . . . . . . 353 18.2 The Converse of the Mean-Value Theorem 358 18.3 Jensen's Inequality . . . . . . . . . . . . . . 360 18.4 Conditional Expected Value Given a Random Variable. 363 18.5 Conditional Law of Y Given X . . . . . . . . 364 18.6 Computation of Conditional Laws . . . . . . 18.7 Existence of Conditional Laws when G = Rk

369 371

IV Operations on Radon Measures 377

19 /l-Adequate Family of Measures 379 19.1 Induced Radon Measure. . . . . 380 19.2 /l-Dense Families of Compact Sets . 382 19.3 Sums of Radon Measures 385 19.4 /l-Adequate Families 387 19.5 /l-Adapted Pairs . . . . . 391

20 Radon Measures Defined by Densities 397 20.1 Integration with Respect to Induced Measures 397 20.2 Radon Measures with Base /l . . 399 20.3 The Radon-Nikodym Theorem . 402 20.4 Duality of LP Spaces . . . . . . . 405

21 Images of Radon Measures and Product Measures 408 21.1 Images of Radon Measures ... . . . . 408 21.2 Decomposition of a Measure in Slices . 410 21.3 Product of Radon Measures. . . . 411

22 Operations on Regular Measures 416 22.1 Some Operations on Regular Measures. . 416 22.2 Baire Sets. . . . . . . . . . . . 419 22.3 Product of Regular Measures . 421 22.4 Change of Variable Formula. . 423

23 Haar Measures 425 23.1 Invariant Measures . . . . . . . . . . . . . . . . . . 426 23.2 Existence and Uniqueness of Left Haar Measure . 429 23.3 Modular Function on G . . . . . . . . . . 433 23.4 Relatively Invariant Measures on a Group 434 23.5 Homogeneous Spaces. . . . . . . . . . . . 436

24

Index

Contents xiii

23.6 Integration with Respect to >.~ . . . . . . . . . . . 442 23.7 Reconstitution of >.~ /3 ....•......•.... • 444 23.8 Quasi-Invariant Measures on Homogeneous Spaces . 446 23.9 Relatively Invariant Measures on G / H . . 449 23.10 Haar Measure on SO(n + 1, R) . 450 23.11 Haar Measure on SH(n, R) . 454

Convolution of Measures 24.1 Convolvable Measures . . . . . . . . . . . . . . . . . . 24.2 Convolution of a Measure and a Function ...... . 24.3 Convolution of a Measure and a Continuous Function 24.4 Convolution of f.L E M(G, C) and f E O(f3) . 24.5 Convolution and Transposition . . . . 24.6 Convolution of Functions on a Group 24.7 Regularization . . . . . . . 24.8 Definition of Gelfand Pair . . . . . . .

465 · 466 · 467 .471 .472 · 476 .479

483 · 488

499

Symbol Index 505