实分析和概率论

媒体关注与评论

　　本书在两个方面获得了极佳的成功。一是它是一本全面、新颖的实分析教程，二是它是一本数学理论完整和自成体系的概率论教程。本书无疑给出了一种严谨和完整的新标准。　　——美国数学会公报　　这是一本非凡的著作。在教学和参考两个方面本书将成为一本标准化教材，它全面地介绍了实分析的必备知识，且证明贯穿全书。一些主题和证明极少在其他教科书中见到。　　——爱丁堡数学会学报　　严谨，精深，新颖，这是一本适用于数学专业研究生的教材。　　——ISI的简短书评

内容概要

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书籍目录

Preface to the Cambridge Edition1 Foundations; Set Theory1.1 Definitions for Set Theory and the Real Number System1.2 Relations and Orderings* 1.3 Transfinite Induction and Recursion1.4 Cardinality1.5 The Axiom of Choice and Its Equivalents2 General Topology2.1 Topologies， Metrics， and Continuity2.2 Compactness and Product Topologies2.3 Complete and Compact Metric Spaces2.4 Some Metrics for Function Spaces2.5 Completion and Completeness of Metric Spaces*2.6 Extension of Continuous Functions*2.7 Uniformities and Uniform Spaces*2.8 Compactification3 Measures3.1 Introduction to Measures3.2 Semirings and Rings3.3 Completion of Measures3.4 Lebesgue Measure and Nonmeasurable Sets*3.5 Atomic and Nonatomic Measures4 Integration4.1 Simple Functions*4.2 Measurability4.3 Convergence Theorems for Integrals4.4 Product Measures*4.5 Daniell-Stone Integrals5 Lp Spaces; Introduction to Functional Analysis5.1 Inequalities for Integrals5.2 Norms and Completeness of LP5.3 Hilbert Spaces5.40rthonormal Sets and Bases5.5 LinearForms on Hilbert Spaces， Inclusions of LP Spaces，and Relations Between Two Measures5.6 Signed Measures6 Convex Sets and Duality of Normed Spaces6.1 Lipschitz， Continuous， and Bounded Functionals6.2 Convex Sets and Their Separation6.3 Convex Functions*6.4 Duality of Lp Spaces6.5 Uniform Boundedness and Closed Graphs*6.6 The Bmnn-Minkowski Inequality7 Measure， Topology， and Differentiation，7.1 Baire and Borel o-Algebras and Regularity of Measures*7.2 Lebesgues Differentiation Theorems*7.3 The Regularity Extension*7.4 The Dual of C(K） and Fourier Series*7.5 Almost Uniform Convergence and Lusins Theorem8 Introduction to Probability Theory8.1 Basic Definitions8.2 Infinite Products of Probability Spaces8.3 Laws of Large Numbers*8.4 Ergodic Theorems9 Convergence of Laws and Central Limit Theorems9.1 Distribution Functions and Densities9.2 Convergence of Random Variables9.3 Convergence of Laws9.4 Characteristic Functions9.5 Uniqueness of Characteristic Functionsand a Central Limit Theorem9.6 Triangular Arrays and Lindebergs Theorem9.7 Sums of Independent Real Random Variables*9.8 The Levy Continuity Theorem; Infinitely Divisibleand Stable Laws10 Conditional Expectations and Martingales10.1 Conditional Expectations10.2 Regular Conditional Probabilities and JensensInequality10.3 Martingales10.4 Optional Stopping and Uniform Integrability10.5 Convergence of Martingales and Submartingales* 10.6 Reversed Martingales and Submartingales* 10.7 Subadditive and Superadditive Ergodic Theorems11 Convergence of Laws on Separable Metric Spaces11.1 Laws and Their Convergence11.2 Lipschitz Functions11.3 Metrics for Convergence of Laws11.4 Convergence of Empirical Measures11.5 Tightness and Uniform Tightness*11.6 Strassens Theorem: Nearby VariablesWith Nearby Laws* 11.7 A Uniformity for Laws and Almost Surely ConvergingRealizations of Converging Laws* 11.8 Kantorovich-Rubinstein Theorems* 11.9 U-Statistics12 Stochastic Processes12.1 Existence of Processes and Brownian Motion12.2 The Strong Markov Property of Brownian Motion12.3 Reflection Principles， The Brownian Bridge，and Laws of Suprema12.4 Laws of Brownian Motion at Markov Times:Skorohod Imbedding12.5 Laws of the Iterated Logarithm13 Measurability: Borel Isomorphism and Analytic Sets* 13.1 Borel Isomorphism* 13.2 Analytic SetsAppendix A Axiomatic Set TheoryA.1 Mathematical LogicA.2 Axioms for Set TheoryA.3 Ordinals and CardinalsA.4 From Sets to NumbersAppendix B Complex Numbers， Vector Spaces，and Taylors Theorem with RemainderAppendix C The Problem of MeasureAppendix D Rearranging Sums of Nonnegative TermsAppendix E Pathologies of Compact Nonmetric SpacesAuthor IndexSubject IndexNotation Index

编辑推荐

　　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一魏尔斯特拉斯定理；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。

作者简介

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。

图书封面

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