本书介绍了现代概率论的基础知识，从有限和可数样本空间的概率论开始，逐步过渡到简单讲授测度论，随后介绍了概率论的一些初步应用，包括独立性和条件期望。书的后半部分涉及高斯随机变量、马尔可夫链、一些连续参数过程（包括布朗运动）以及最后的离散和连续参数的鞅。本书是关于概率论及其所需测度论的一本自封的入门书籍。

Preface
Chapter 1. Some Background and Preliminaries
1.1. The Language of Probability Theory
1.1.1. Sample Spaces and Events
1.1.2. Probability Measures
Exercises for 1.1
1.2. Finite and Countable Sample Spaces
1.2.1. Probability Theory on a Countable Space
1.2.2. Uniform Probabilities and Coin Tossing
1.2.3. Tournaments
1.2.4. Symmetric Random Walk
1.2.5. De Moivre's Central Limit Theorem
1.2.6. Independent Events
1.2.7. The Arc Sine Law
1.2.8. Conditional Probability
Exercises for 1.2
1.3. Some Non-Uniform Probability Measures
1.3.1. Random Variables and Their Distributions
1.3.2. Biased Coins
1.3.3. Recurrence and Transience of Random Walks
Exercises for 1.3
1.4. Expectation Values
1.4.1. Some Elementary Examples
1.4.2. Independence and Moment Generating Functions
1.4.3. Basic Convergence Results
Exercises for 1.4
Comments on Chapter 1
Chapter 2. Probability Theory on Uncountable Sample Spaces
2.1. A Little Measure Theory
2.1.1. Sigma Algebras, Measurable Functions, and Measures
2.1.2. Ⅱ-and A-Systems
Exercises for 2.1
2.2. A Construction of Pp on {0，1}z+
2.2.1. The Metric Space {0，1}z+
2.2.2. The Construction
Exercises for 2.2
2.3. Other Probability Measures
2.3.1. The Uniform Probability Measure on [0，1]
2.3.2. Lebesgue Measure on R
2.3.3. Distribution Functions and Probability Measures
Exercises for 2.3
2.4. Lebesgue Integration
2.4.1. Integration of Functions
2.4.2. Some Properties of the Lebesgue Integral
2.4.3. Basic Convergence Theorems
2.4.4. Inequalities
2.4.5. Fubini's Theorem
Exercises for 2.4
2.5. Lebesgue Measure on RN
2.5.1. Polar Coordinates
2.5.2. Gaussian Computations and Stirling's Formula
Exercises for 2.5
Comments on Chapter 2
Chapter 3. Some Applications to Probability Theory
3.1. Independence and Conditioning
3.1.1. Independent σ-Algebras
3.1.2. Independent Random Variables
3.1.3. Conditioning
3.1.4. Some Properties of Conditional Expectations
Exercises for 3.1
3.2. Distributions that Admit a Density
3.2.1. Densities
3.2.2. Densities and Conditioning
Exercises for 3.2
3.3. Summing Independent Random Variables
3.3.1. Convolution of Distributions
3.3.2. Some Important Examples
3.3.3. Kolmogorov's Inequality and the Strong Law
Exercises for 3.3
Comments on Chapter 3
Chapter 4. The Central Limit Theorem and Gaussian Distributions
4.1. The Central Limit Theorem
4.1.1. Lindeberg's Theorem
Exercises for 4.1
4.2. Families of Normal Random Variables
4.2.1. Multidimensional Gaussian Distributions
4.2.2. Standard Normal Random Variables
4.2.3. More General Normal Random Variables
4.2.4. A Concentration Property of Gaussian Distributions
4.2.5. Linear Transformations of Normal Random Variables
4.2.6. Gaussian Families
Exercises for 4.2
Comments on Chapter 4
Chapter 5. Discrete Parameter Stochastic Processes
5.1. Random Walks Revisited
5.1.1. Immediate Rewards
5.1.2. Computations via Conditioning
Exercises for 5.1
5.2. Processes with the Markov Property
5.2.1. Sequences of Dependent Random Variables
5.2.2. Markov Chains
5.2.3. Long-Time Behavior
5.2.4. An Extension
Exercises for 5.2
5.3. Markov Chains on a Countable State Space
5.3.1. The Markov Property
5.3.2. Return Times and the Renewal Equation
5.3.3. A Little Ergodic Theory
Exercises for 5.3
Comments on Chapter 5
Chapter 6. Some Continuous-Time Processes
6.1. Transition Probability Functions and Markov Processes
6.1.1. Transition Probability Functions
Exercises for 6.1
6.2. Markov Chains Run with a Poisson Clock
6.2.1. The Simple