严格概率论初阶FIRST LOOK AT RIGOROUS PROBABILITY THEORY

This textbook is an introduction to rigorous probability theory using measure theory. It provides rigorous, complete proofs of all the essential introductory mathematical results of probability theory and measure theory. More advanced or specialised areas are only hinted at. For example, the text includes a complete proof of the classical central limit theorem, including the necessary continuity theorem for characteristic functions, but the more general Lindeberg central limit theorem is only outlined and is not proved. Similarly, all necessary facts from measure theory are proved before they are used, but more abstract or advanced measure theory results are not included. Furthermore, measure theory is discussed as much as possible purely in terms of probability, as opposed to being treated as a separate subject which must be mastered before probability theory can be understood.

Preface
1 The need for measure theory
1.1 Various kinds of random variables
1.2 The uniform distribution and non-measurable sets
　1.3 Additional exercises
　1.4 Section summary
2 Probability triples
　2.1 Basic definition
　2.2 Discrete probability spaces
　2.3 Constructing Lebesgue measure .
　2.4 The extension theorem
　2.5 More on Lebesgue measure
　2.6 Coin tossing and other measures
　2.7 Additional exercises
　2.8 Section summary
3 Further probabilistic foundations
　3.1 Random variables
　3.2 Independence
　3.3 Continuity of probabilities
　3.4 Limit events
　3.5 Tail fields
　3.6 Additional exercises
　3.7 Section summary
4 Expected values
　4.1 Simple random variables
　4.2 General rion-negative random variables . .
　4.3 Arbitrary random variables
　4.4 The integration connection
　4.5 Additional exercises
　4.6 Section summary
5 Inequalities and laws of large numbers
5.1 Weak law of large numbers
5.2 Strong law of large numbers
5.3 Eliminating the moment conditions
5.4 Additional exercises
5.5 Section summary
6 Distributions of random variables
6.1 Change of variable theorem
6.2 Examples of distributions
6.3 Additional exercises
6.4 Section summary
7 Stochastic processes and gambling games
7.1 A first existence theorem
7.2 Gambling and gambler's ruin
7.3 Gambling policies
7.4 Additional exercises
7.5 Section summary
8 Discrete Markov chains
8.1 A Markov chain existence theorem
8.2 Transience, recurrence, and irreducibility
8.3 Stationary distributions and convergence
8.4 Existence of stationary distributions
8.5 Additional exercises
8.6 Section summary
9 Some further probability results
9.1 Limit theorems
9.2 Differentiation and expectation
9.3 Moment generating functions and large deviations
9.4 Convolution and Fubini's Theorem
9.5 Additional exercises
9.6 Section summary
10 Weak convergence
10.1 Additional exercises
10.2 Section summary
11 Chareateristic functions
12 Decomposition of probability laws
13 Conditional probability and expectation
14 Martinagles
15 Introduction to other stochastic processes
Appendix:Mathematical Background
Bibligraphy
Index