实分析和概率论(英文版.第2版)

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-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 Toplogies,Metrics,and Continuity 2.2 Compactness and Product Toplogies 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 …… 6. Convex sets and duality of normed spaces 7. Measure, topology, and differentiation 8. Introduction to probability theory 9. Convergence of laws and central limit theorems 10. Conditional expectations and martingales 11. Convergence of laws on separable metric spaces 12. Stochastic processes 13. Measurability: Borel isomorphism and analytic sets Appendixes: A. Axiomatic set theory Appendixes: B. Complex numbers, vector spaces, and Taylor’s theorem with remainder Appendixes: C. The problem of measure Appendixes: D. Rearranging sums of Nonnegative terms Appendixes: E. Pathologies of compact Nonmetric spaces Author Index Subject Index Notation Index