1 Foundations;Set Theory

1.1 Definitions for Set Theory and the Real Number System

1.2 Relations and Orderings

1.3 Transfinite Induction and Recursion

1.4 Cardinality

1.5 The Axiom of Choice and Its Equivalents

2 General Topology

2.1 Topologies,Metrics,and Continuity

2.2 Compactness and Product Topologies

2.3 Complete and Compact Metric Spaces

2.4 Some Metrics for Function Spaces

2.5 Completion and Completeness of Metric Spaces

2.6 Extension of Continuous Functions

2.7 Uniformities and Uniform Spaces

2.8 Compactification

3 Measures

3.1 Introduction to Measures

3.2 Semirings and Rings

3.3 Completion of Measures

3.4 Lebesgue Measure and Nonmeasurable Sets

3.5 Atomic and Nonatomic Measures

4 Integration

4.1 Simple Functions

4.2 Measurability

4.3 Convergence Theorems for Integrals

4.4 Product Measures

4.5 Daniell-Stone Integrals

5 Lp Spaces;Introduction to Functional Analysis

5.1 Inequalities for Integrals

5.2 Norms and Completeness of Lp

5.3 Hilbert Spaces

5.4Orthonormal Sets and Bases

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

5.6 Signed Measures

6 Convex Sets and Duality of Normed Spaces

6.1 Lipschitz,Continuous,and Bounded Functionals

6.2 Convex Sets and Their Separation

6.3 Convex Functions

6.4 Duality of Lp Spaces

6.5 Uniform Boundedness and Closed Graphs

6.6 The Brunn-Minkowski Inequality

7 Measure,Topology,and Differentiation

7.1 Baire and Borel σ-Algebras and Regularity of Measures

7.2 Lebesgue's Differentiation Theorems

7.3 The Regularity Extension

7.4 The Dual of C(K)and Fourier Series

7.5 Almost Uniform Convergence and Lusin's Theorem

8 Introduction to Probability Theory

8.1 Basic Definitions

8.2 Infinite Products of Probability Spaces

8.3 Laws of Large Numbers

8.4 Ergodic Theorems

9 Convergence of Laws and Central Limit Theorems

9.1 Distribution Functions and Densities

9.2 Convergence of Random Variables

9.3 Convergence of Laws

9.4 Characteristic Functions

9.5 Uniqueness of Characteristic Functions and a Central Limit Theorem

9.6 Triangular Arrays and Lindeberg's Theorem

9.7 Sums of Independent Real Random Variables

9.8 The Lévy Continuity Theorem;Infinitely Divisible and Stable Laws

10 Conditional Expectations and Martingales

10.1 Conditional Expectations

10.2 Regular Conditional Probabilities and Jensen's Inequality

10.3 Martingales

10.4 Optional Stopping and Uniform Integrability

10.5 Convergence of Martingales and Submartingales

10.6 Reversed Martingales and Submartingales

10.7 Subadditive and Superadditive Ergodic Theorems

11 Convergence of Laws on Separable Metric Spaces

11.1 Laws and Their Convergence

11.2 Lipschitz Functions

11.3 Metrics for Convergence of Laws

11.4 Convergence of Empirical Measures

11.5 Tightness and Uniform Tightness

11.6 Strassen's Theorem:Nearby Variables with Nearby Laws

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

11.8 Kantorovich-Rubinstein Theorems

11.9 U-Statistics

12 Stochastic Processes

12.1 Existence of Processes and Brownian Motion

12.2 The Strong Markov Property of Brownian Motion

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

12.4 Laws of Brownian Motion at Markov Times:Skorohod Imbedding

12.5 Laws of the Iterated Logarithm

13 Measurability:Borel Isomorphism and Analytic Sets

13.1 Borel Isomorphism

13.2 Analytic Sets

Appendix A Axiomatic Set Theory

A.1 Mathematical Logic

A.2 Axioms for Set Theory

A.3 Ordinals and Cardinals

A.4 From Sets to Numbers

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

Appendix C The Problem of Measure

Appendix D Rearranging Sums of Nonnegative Terms

Appendix E Pathologies of Compact Nonmetric Spaces

Author Index

Subject Index

Notation Index