这是一本广受称赞的教科书，清晰地讲解了现代概率论以及度量空间与概率测度之间的相互作用。本书分两部分，第一部分介绍了实分析的内容，包括基本集合论、一般拓扑学、测度论、积分法、巴拿赫空间和拓扑空间中的泛函分析导论、凸集和函数、拓扑空间上的测度等。第二部分介绍了基于测度论的概率方面的内容，包括大数律、遍历定理、中心极限定理、条件期望、鞅收敛等。另外，随机过程一章(第12章)还介绍了布朗运动和布朗桥。与前版相比，本版内容更完善，一开始就介绍了实数系的基础和泛代数中的一致逼近的斯通一魏尔斯特拉斯定理；修订和改进了几节的内容，扩充了大量历史注记；增加了很多新的习题，以及对一些习题的解答的提示。

Preface to the Cambridge Edition

  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

 5.40rthonormal 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 Bmnn-Minkowski Inequality

7 Measure, Topology, and Differentiation,

 7.1 Baire and Borel o'-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 Levy 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