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 Inequali