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本书是《概率与测度》第3版，新版保留了原先的风格，将测度论和概率论有机结合在一起，把相关内容混合排列。概率问题会引起学生学习测度论的兴趣，而测度论知识又反过来应用到概率论中。本书主要内容包括概率、测度、积分、随机变量及数学期望、分布的收敛问题、导数与条件期望，随机过程等。本版改进了布朗运动的叙述方式，并以遍历理论代替排队论。本书的读者对象为高年级学生、科研人员和工程技术人员，对数学、统计、经济等相前专业的学生尤其适用。

FOREWORD

PREFACE

Patrick Billingsley 1925- 2011

Chapter 1

PROBABILITY

1. BOREL'S NORMAL NUMBER THEOREM, 1

The Unit Interval——The Weak Law of Large

Numbers——The Strong Law of Large Numbers——Strong Law

Versus Weak—— Length——The Measure Theory of Diophantine

Approximation*

2. PROBABILITY MEASURES, 18

Spaces ——Assigning Probabilities——Classes of Sets——Probability

Measures——Lebesgue Measure on the Unit Interval——Sequence

Space* - Constructing σ-Fields*

3. EXISTENCE AND EXTENSION, 39

Construction of the Extension——Uniqueness and the π-λ

Theorem——Monotone Classes——Lebesgue Measure on the Unit

Interval- Completeness—— Nonmeasurable Sets——Two

Impossibility Theorems*

4. DENUMERABLE PROBABILITIES, 53

General Formulas—— Limit Sets-Independent

Events——Subfields——The Borel-Cantelelli

Lemmas——The Zero-One Law

5. SIMPLE RANDOM VARIABLES, 72

Definition—— Convergence of Random

Variables——Independence——Existence of Independent

Sequences—— Expected Value——Inequalities

6. THE LAW OF LARGE NUMBERS, 90

The Strong Law——The Weak Law——Bernstein's

Theorem——A Refinement of the Second BoreI-Cantelli

Lemma

7. GAMBLING SYSTEMS, 98

Gambler's Ruin——Selection Systems——Gambling Policies——Bold

Play*——Timid Play*

8. MARKOVCHAINS, 117

Definitions—— Higher-Order Transitions ——An Existence

Theorem——Transience and Persistence——Another Criterion for

Persistence——Stationary Distributions—— Exponential

Convergence*——Optimal Stopping*

9. LARGE DEVIATIONS AND THE LAW

OF THE ITERATED LOGARITHM, 154

Moment Generating Functions——Large Deviations —— Chernoff's

Theorem*——The Law of the Iterated Logarithm

Chapter 2

MEASURE 167

10. GENERAL MEASURES, 167

Classes of Sets—— Conventions Involving

∞ —— Measures—— Uniqueness

11. OUTER MEASURE, 174

Outer Measure——Extension——An Approximation Theorem

12. MEASURES IN EUCLIDEAN SPACE, 181

Lebesgue Measure——Regularity——Specifying Measures on the

Line——Specifying Measures in Rk-strange Euclidean Sets*

13. MEASURABLE FUNCTIONS AND MAPPINGS, 192

Measurable Mappings—— Mappings into Rk- Limits and

Measurability——Transformations of Measures

14. DISTRIBUTION FUNCTIONS, 198

Distribution Functions——Exponential Distributions——Weak

Convergence—— Convergence of Types* —— Extremal

Distributions*

Chapter 3

INTEGRATION 211

15. THE INTEGRAL, 211

Definition —— Nonnegative Functions—— Uniqueness

16. PROPERTIES OF THE INTEGRAL, 218

Equalities and Inequalities——Integration to the Limit——Integration

over Sets—— Densities—— Change of Variable—— Uniform

Integrability—— Complex Functions

17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE, 234

The Lebesgue Integral on the Line——The Riemann

Integral——The Fundamental Theorem of Calculus——Change of

Variable——The Lebesgue Integral in Rk——Stieltjes Integrals

18. PRODUCT MEASURE AND FUBINI'S THEOREM, 245

Product Spaces—— Product Measure—— Fubini's

Theorem——Integration by Parts—— Products of Higher Order

19. THE Lp SPACES*, 256

Definitions—— Completeness and Separability—— Conjugate

Spaces——Weak Compactness——Some Decision

Theory——The Space L2-An Estimation Problem

Chapter 4

RANDOM VARIABLES AND EXPECTED VALUES 271

20. RANDOM VARIABLES AND DISTRIBUTIONS, 271

Random Variables and Vectors——

Subfields—— Distributions —— Multidimensional

Distributions——Independence——Sequences of Random

Variables——Convolution——Convergence in

Probability——The Glivenko-Cantelli Theorem*

21. EXPECTED VALUES, 291

Expected Value as Integral——Expected Values

and Limits—— Expected Values and

Distributions—— Moments——Inequalities——Joint

Integrals——Independence and Expected Value—— Moment

Generating Functions

22. SUMS OF INDEPENDENT RANDOM VARIABLES, 300

The Strong Law of Large Numbers——The Weak Law

and Moment Generating Functions——Kolmogorov's Zero-One

Law—— Maximal Inequalities—— Convergence of Random

Series——Random Taylor Series*

23. THE POISSON PROCESS, 316

Characterization of the Exponential Distribution——The Poisson

Process——The Poisson Approximation——Other Characterizations

of the Poisson Process——Stochastic

Processes

24. THE ERGODIC THEOREM*, 330

Measure-Preserving Transformations—— Ergodicity—— Ergodicity of

Rotations——Proof of the Ergodic Theorem——The

Continued-Fraction Transformation—— Diophantine

Approximation

Chapter 5

CONVERGENCE OF DISTRIBUTIONS 349

25. WEAK CONVERGENCE, 349

Definitions—— Uniform Distribution Modulo 1 * ——Convergence

in Distribution——Convergence in Probability——Fundamental

Theorems——Helly's Theorem——Integration to the Limit

26. CHARACTERISTIC FUNCTIONS, 365

Definition —— Moments and Derivatives——

Independence——Inversion and the Uniqueness Theorem——The

Continuity Theorem—— Fourier Series*

27. THE CENTRAL LIMIT THEOREM, 380

Identically Distributed Summands——The Lindeberg

and Lyapounov Theorems——Dependent Variables*

28. INFINITELY DIVISIBLE DISTRIBUTIONS*, 394

Vague Convergence——The Possible Limits——Characterizing

the Limit

29. LIMIT THEOREMS IN Rk, 402

The Basic Theorems—— Characteristic Functions—— Normal

Distributions in Rk——The Central Limit Theorem

30. THE METHOD OF MOMENTS*, 412

The Moment Problem——Moment Generating Functions——Central

Limit Theorem by Moments——Application

to Sampling Theory——Application to Number Theory

Chapter 6

DERIVATIVES AND CONDITIONAL PROBABILITY 425

31. DERIVATIVES ON THE LINE*, 425

The Fundamental Theorem of Calculus——Derivatives

of Integrals——Singular Functions——Integrals

of Derivatives——Functions of Bounded Variation

32. THE RADON-NIKODYM THEOREM, 446

Additive Set Functions——The Hahn Decomposition——Absolute

Continuity and Singularity——The Main Theorem

33. CONDITIONAL PROBABILITY, 454

The Discrete Case——The General Case——Properties

of Conditional Probability—— Difficulties and

Curiosities—— Conditional Probability Distributions

34. CONDITIONAL EXPECTATION, 472

Definition—— Properties of Conditional Expectation——Conditional

Distributions and Expectations—— Sufficient

Subfields* —— Minimum-Variance Estimation*

35. MARTINGALES, 487

Definition —— Su bmartingales—— Gambling —— Functions

of Martingales—— Stopping Times—— Inequalities—— Convergence

Theorems——Applications: Derivatives—— Likelihood

Ratios—— Reversed Martingales——Applications: de Finetti's

Theorem——Bayes Estimation——A Central Limit Theorem*

Chapter 7

STOCHASTIC PROCESSES 513

36. KOLMOGOROV'S EXISTENCE THEOREM, 513

Stochastic Processes—— Finite-Dimensional

Distributions—— Product Spaces—— Kolmogorov's Existence

Theorem——The Inadequacy of RT-A Return to Ergodic

Theory——The Hewitt-Savage Theorem*

37. BROWNIAN MOTION, 530

Definition ——Continuity of Paths—— Measurable

Processes——Irregularity of Brownian Motion Paths——The Strong

Markov Property——The Reflection Principle——Skorohod

Embedding ——I nvariance*

38. NONDENUMERABLE PROBABILITIES, 558

Introduction —— Definitions—— Existence

Theorems——Consequences of Separability*

APPENDIX

NOTES ON THE PROBLEMS

BIBLIOGRAPHY

INDEX

Patrick Billingsley是芝加哥大学统计学和数学教授。他是《管理和经济类适用的统计学》（与Watson等人合作）、《统计推断要义》（与D.L.Huntsberger合作）、《概率测度的收敛性》等书的作者，曾任美国数理统计学会的《概率年刊》主编，他从普林斯顿大学获得哲学博士。

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书籍介绍

本书是《概率与测度》第3版，新版保留了原先的风格，将测度论和概率论有机结合在一起，把相关内容混合排列。概率问题会引起学生学习测度论的兴趣，而测度论知识又反过来应用到概率论中。本书主要内容包括概率、测度、积分、随机变量及数学期望、分布的收敛问题、导数与条件期望，随机过程等。本版改进了布朗运动的叙述方式，并以遍历理论代替排队论。本书的读者对象为高年级学生、科研人员和工程技术人员，对数学、统计、经济等相前专业的学生尤其适用。