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1 Foundations;Set Theory 1

1.1 Definitions for Set Theory and the Real Number System 1

1.2 Relations and Orderings 9

1.3 Transfinite Induction and Recursion 12

1.4 Cardinality 16

1.5 The Axiom of Choice and Its Equivalents 18

2 General Topology 24

2.1 Topologies,Metrics,and Continuity 24

2.2 Compactness and Product Topologies 34

2.3 Complete and Compact Metric Spaces 44

2.4 Some Metrics for Function Spaces 48

2.5 Completion and Completeness of Metric Spaces 58

2.6 Extension of Continuous Functions 63

2.7 Uniformities and Uniform Spaces 67

2.8 Compactification 71

3 Measures 85

3.1 Introduction to Measures 85

3.2 Semirings and Rings 94

3.3 Completion of Measures 101

3.4 Lebesgue Measure and Nonmeasurable Sets 105

3.5 Atomic and Nonatomic Measures 109

4 Integration 114

4.1 Simple Functions 114

4.2 Measurability 123

4.3 Convergence Theorems for Integrals 130

4.4 Product Measures 134

4.5 Daniell-Stone Integrals 142

5 Lp Spaces;Introduction to Functional Analysis 152

5.1 Inequalities for Integrals 152

5.2 Norms and Completeness of Lp 158

5.3 Hilbert Spaces 160

5.4Orthonormal Sets and Bases 165

5.5 LinearForms on Hilbert Spaces,Inclusions of Lp Spaces,and Relations Between Two Measures 173

5.6 Signed Measures 178

6 Convex Sets and Duality of Normed Spaces 188

6.1 Lipschitz,Continuous,and Bounded Functionals 188

6.2 Convex Sets and Their Separation 195

6.3 Convex Functions 203

6.4 Duality of Lp Spaces 208

6.5 Uniform Boundedness and Closed Graphs 211

6.6 The Brunn-Minkowski Inequality 215

7 Measure,Topology,and Differentiation 222

7.1 Baire and Borel σ-Algebras and Regularity of Measures 222

7.2 Lebesgue's Differentiation Theorems 228

7.3 The Regularity Extension 235

7.4 The Dual of C(K)and Fourier Series 239

7.5 Almost Uniform Convergence and Lusin's Theorem 243

8 Introduction to Probability Theory 250

8.1 Basic Definitions 251

8.2 Infinite Products of Probability Spaces 255

8.3 Laws of Large Numbers 260

8.4 Ergodic Theorems 267

9 Convergence of Laws and Central Limit Theorems 282

9.1 Distribution Functions and Densities 282

9.2 Convergence of Random Variables 287

9.3 Convergence of Laws 291

9.4 Characteristic Functions 298

9.5 Uniqueness of Characteristic Functions and a Central Limit Theorem 303

9.6 Triangular Arrays and Lindeberg's Theorem 315

9.7 Sums of Independent Real Random Variables 320

9.8 The Lévy Continuity Theorem;Infinitely Divisible and Stable Laws 325

10 Conditional Expectations and Martingales 336

10.1 Conditional Expectations 336

10.2 Regular Conditional Probabilities and Jensen's Inequality 341

10.3 Martingales 353

10.4 Optional Stopping and Uniform Integrability 358

10.5 Convergence of Martingales and Submartingales 364

10.6 Reversed Martingales and Submartingales 370

10.7 Subadditive and Superadditive Ergodic Theorems 374

11 Convergence of Laws on Separable Metric Spaces 385

11.1 Laws and Their Convergence 385

11.2 Lipschitz Functions 390

11.3 Metrics for Convergence of Laws 393

11.4 Convergence of Empirical Measures 399

11.5 Tightness and Uniform Tightness 402

11.6 Strassen's Theorem:Nearby Variables with Nearby Laws 406

11.7 A Uniformity for Laws and Almost Surely Converging Realizations of Converging Laws 413

11.8 Kantorovich-Rubinstein Theorems 420

11.9 U-Statistics 426

12 Stochastic Processes 439

12.1 Existence of Processes and Brownian Motion 439

12.2 The Strong Markov Property of Brownian Motion 450

12.3 Reflection Principles,The Brownian Bridge,and Laws of Suprema 459

12.4 Laws of Brownian Motion at Markov Times:Skorohod Imbedding 469

12.5 Laws of the Iterated Logarithm 476

13 Measurability:Borel Isomorphism and Analytic Sets 487

13.1 Borel Isomorphism 487

13.2 Analytic Sets 493

Appendix A Axiomatic Set Theory 503

A.1 Mathematical Logic 503

A.2 Axioms for Set Theory 505

A.3 Ordinals and Cardinals 510

A.4 From Sets to Numbers 515

Appendix B Complex Numbers,Vector Spaces,and Taylor's Theorem with Remainder 521

Appendix C The Problem of Measure 526

Appendix D Rearranging Sums of Nonnegative Terms 528

Appendix E Pathologies of Compact Nonmetric Spaces 530

Author Index 541

Subject Index 546

Notation Index 554