REAL ANALYSIS AND PROBABI LITY

Richard M. Dudley Massachusetts Institute of Technology

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Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California

Contents

C H A P T E R 1

Foundations; Set Theory

1.1 Definitions for Set Theory 1 1.2 Relations and Orderings 7

*1.3 Transfinite Induction and Recursion 9 1.4 Cardinality 12 1.5 The Axiom of Choice and Its Equivalents 14

C H A P T E R 2

General Topology 19

2.1 Topologies, Metrics, and Continuity 19 2.2 Compactness and Product Topologies 27 2.3 Complete and Compact Metrie Spaces 34 2.4 Some Metrics for Function Spaces 37 2.5 Completion and Completeness of

Metrie Spaces 43 *2.6 Extension of Continuous Functions 47 *2.7 Uniformities and Uniform Spaces 50 *2.8 Compactification 53

VII

C H A P T E R 3

Measures 63

3.1 Introduction to Measures 63 3.2 Semirings and Rings 70 3.3 Completion of Measures 75 3.4 Lebesgue Measure and Nonmeasurable Sets 79

*3.5 Atomic and Nonatomic Measures 82

C H A P T E R 4

Integration 86

4.1 Simple Functions 86 *4.2 Measurability 94 4.3 Convergence Theorems for Integrals 99 4.4 Product Measures 102

*4.5 Daniell-Stone Integrals 108

C H A P T E R 5

W Spaces; Introduction to Functional Analysis 116

5.1 Inequalities for Integrals 116 5.2 Norms and Completeness of Lp 120 5.3 Hubert Spaces 123 5.4 Orthonormal Sets and Bases 126 5.5 Linear Forms on Hubert Spaces, Inclusions of Lp Spaces

and Relations Between Two Measures 133 5.6 Signed Measures 136

C H A P T E R 6

Convex Sets and Duality of Normed Spaces 145

6.1 Lipschitz, Continuous, and Bounded Functionals 145 6.2 Convex Sets and Their Separation 150 6.3 Convex Functions 157

Contents ix

*6.4 DualityofZ/ Spaces 161 6.5 Uniform Boundedness and Closed Graphs 164

*6.6 The Brunn-Minkowski Inequality 167

C H A P T E R 7

Measure, Topology, and Differentiation 173

7.1 Baire and Borel «r-Algebras and Regularity ofMeasures 173

*7.2 Lebesgue's Differentiation Theorems 177 *7.3 The Regularity Extension 183 *7.4 The Dual of C(K) and Fourier Series 186 *7.5 Almost Uniform Convergence and Lusin's Theorem 189

C H A P T E R 8

Introduction to Probability Theory

8.1 Basic Definitions 196 8.2 Infinite Products of Probability Spaces 8.3 Laws of Large Numbers 204

*8.4 Ergodic Theorems 209

C H A P T E R 9

Convergence of Laws and Central Limit Theorems 221

9.1 Distribution Functions and Densities 221 9.2 Convergence of Random Variables 225 9.3 Convergence of Laws 228 9.4 Characteristic Functions 233 9.5 Uniqueness of Characteristic Functions and a Central

Limit Theorem 237 9.6 Triangulär Arrays and Lindeberg's Theorem 247 9.7 Sums of Independent Real Random Variables 250

*9.8 The Levy Continuity Theorem; Infinitely Divisible and Stable Laws 255

195

199

C H A P T E R 10

Conditional Expectations and Martingales 264

10.1 Conditional Expectations 264 10.2 Regulär Conditional Probabilities and Jensen's Inequality 268 10.3 Martingales 276 10.4 Optional Stopping and Uniform Integrability 280 10.5 Convergence of Martingales and Submartingales 285

*10.6 Reversed Martingales and Submartingales 289 *10.7 Subadditive and Superadditive Ergodic Theorems 292

C H A P T E R 11

Convergence of Laws on Separable Metrie Spaces 302

11.1 Laws and Their Convergence 302 11.2 Lipschitz Functions 306 11.3 Metrics for Convergence of Laws 309 11.4 Convergence of Empirical Measures 313 11.5 Tightness and Uniform Tightness 315

*11.6 Strassen's Theorem: Nearby Variables With Nearby Laws 318 *11.7 A Uniformity for Laws and Almost Surely Converging

Realizations of Converging Laws 324 *11.8 Kantorovich-Rubinstein Theorems 329 *11.9 t/-Statistics 334

C H A P T E R 12

Stochastic Processes 346

12.1 Existence of Processes and Brownian Motion 346 12.2 The Strong Markov Property of Brownian Motion 354 12.3 Reflection Principles, The Brownian Bridge, and

Laws of Suprema 360 12.4 Laws of Brownian Motion at Stopping Times:

Skorohod Imbedding 368 12.5 Laws of The Iterated Logarithm 374

* C H A P T E R 13

Contents xi

Measurability: Borel Isomorphism and Analytic Sets 383

*13.1 Borel Isomorphism 383 *13.2 Analytic Sets 388

A P P E N D I X E S

A: Axiomatic Set Theory 396

A.l Mathematical Logic 396 A.2 Axioms for Set Theory 398 A.3 Ordinals and Cardinais 402 A.4 From Sets to Numbers 407

B: Complex Numbers, Vector Spaces, and Taylor's Theorem

with Remainder 411

C: The Problem of Measure 415

D: Rearranging Sums of Nonnegative Terms 417

E: Pathologies of Compact Nonmetric Spaces 419

Author Index 428

Subject Index 431