测度论

章节摘录

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前言

为了更好地借鉴国外数学教育与研究的成功经验，促进我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，本着“为我国热爱数学的青年创造一个较好的学习数学的环境”这一宗旨，天元基金赞助出版“天元基金影印数学丛书”。该丛书主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。为了提高我国数学研究生教学的水平，暂把选书的目标确定在研究生教材上。当然，有的书也可作为高年级本科生教材或参考书，有的书则介于研究生教材与专著之间。欢迎各方专家、读者对本丛书的选题、印刷、销售等工作提出批评和建议。

内容概要

作者：（俄罗斯）博根切维（V.I.Bogachev）

书籍目录

Preface to Volume 2Chapter 6.  Borel, Baire and Souslin sets     6.1.   Metric and topological Spaces     6.2.   Borel sets     6.3.   Baire sets     6.4.   Products of topological spaces     6.5.    Countably generated a-algebras     6.6.    Souslin sets and their separation     6.7.    Sets in Souslin spaceS     6.8.   Mappings of Souslin spaces     6.9.   Measurable choice theorems     6.10.   Supplements and exercises              Borel and Baire sets (43). Souslin setsas projeCtions (46)./C-analytic              and F-analytic sets (49). Blackwell spaces (50). Mappings of Souslin              spaces (51). Measurability in normed spaces (52). The Skorohod              space (53). Exercises (54).Chapter 7.  Measures on topological spaces     7.1.   Borel, Baire and Radon measures     7.2.    T-additive measures     7.3.    Extensions of measures     7.4.   Measures on Souslin spaces     7.5.    Perfect measures     7.6.   Products of measures     7.7.   The Kolmogorov theorem     7.8.   The Daniell integral     7.9.   Measures as functionals     7.10.  The regularity of measures in terms of functionals     7.11.  Measures on locally compact spaces     7.12.  Measures on linear spaces     7.13.  Characteristic functionals     7.14.  Supplements and exercises              Extensions of product measure (126). Measurability on products (129).               Marfk spaces (130). Separable measures (132). Diffused and atomless                measures (133). Completion regular measures (133). Radon                spaces (135). Supports of measures (136). Generalizations of Lusin's                theorem (137). Metric outer measures (140). Capacities (142).                Covariance operators and means of measures (142). The Choquet                representation (145). Convolution (146). Measurable linear                functions (149). Convex measures (149). Pointwise convergence (151).                Infinite Radon measures (154). Exercises (155).Chapter 8.  Weak convergence of measures     8.1.    The definition of weak convergence     8.2.    Weak convergence of nonnegative measures     8.3.    The case of a metric space     8.4.    Some properties of weak convergence     8.5.    The Skorohod representation     8.6.    Weak compactness and the Prohorov theorem     8.7.    Weak sequential completeness     8.8.    Weak convergence and .the Fourier transform     8.9.    Spaces of measures with the weak topology     8.10.   Supplements and exercises                Weak compactness (217). Prohorov spaces (219). The weak sequential                completeness of spaces of measures (226). The A-topology (226).                Continuous mappings of spaces of measures (227). The separability                of spaces of measures (230). Young measures (231). Metrics on                spaces of measures (232). Uniformly distributed sequences (237).                Setwise convergence of measures (241). Stable convergence and                ws-topology (246). ,Exercises (249)Chapter 9.  Transformations of measures and isomorphisms     9.1.    Images and preimages of measures     9.2.    Isomorphisms of measure spaces     9.3.    Isomorphisms of measure algebras     9.4.    Lebesgue-Rohlin spaces     9.5.    Induced point isomorphisms     9.6.   Topologically equivalent measures     9.7.    Continuous images of Lebesgue measure     9.8.    Connections with extensions of measures     9,9.    Absolute continuity of the images of measures     9.10.   Shifts of measures along integral curves     9.11.  Invariant measures and Haar measures     9.12.   Supplements and exercises                Projective systems of measures (308). Extremal preimages of                measures and uniqueness (310). Existence of atomless measures (317).                Invariant and quasi-invariant measures of transformations (318). Point                and Boolean isomorphisms (320). Almost homeomorphisms (323).                Measures with given marginal projections (324). The Stone                representation (325). The Lyapunov theorem (326). Exercises (329)Chapter 10.  Conditional measures and conditional                  expectations     10.1.    Conditional expectations     10.2.    Convergence of conditional expectations     10.3.   Martingales     10.4.   Regular conditional measures     10.5.   Liftings and conditional measures     10.6.    Disintegrations of measures     10.7.   Transition measures     10.8.   Measurable partitions     10.9.   Ergodic theorems     10.10.   Supplements and exercises               Independence (398). Disintegrations (403). Strong liftings (406)                Zero-one laws (407). Laws of large numbers (410). Gibbs               measures (416). Triangular mappings (417). Exercises (427)Bibliographical and Historical CommentsReferencesAuthor IndexSubject Index

编辑推荐

《测度论(第2卷)(影印版)》：天元基金影印数学丛书

作者简介

《测度论(第2卷)(影印版)》是作者在莫斯科国立大学数学力学系的讲稿基础上编写而成的。第二卷介绍测度论的专题性的内容，特别是与概率论和点集拓扑有关的课题：Borel集，Baire集，Souslin集，拓扑空间上的测度，Kolmogorov定理，Daniell积分，测度的弱收敛，Skorohod表示，Prohorov定理，测度空间上的弱拓扑，Lebesgue-Rohlin空间，Haar测度，条件测度与条件期望，遍历理论等。每章最后都附有非常丰富的补充与练习，其中包含许多有用的知识，例如：Skorohod空间，Blackwell空间，Marik空间，Radon空间，推广的Lusin定理，容量，Choquet表示，Prohorov空间，Young测度等。书的最后有详尽的参考文献及历史注记。这是一本很好的研究生教材和教学参考书。

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