简介
本书是天元基金影印数学丛书之一，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。全面丰富地阐述了现代侧度论的基本理论，不要求读者从头到尾系统阅读，特别，补充内容几乎与各章节彼此无关，主要针对那些对测度论有兴趣深入了解的研究生、侧度论和积分理论课程的教师，以及数学各领域的研究人员。全书共分两卷。第二卷介绍侧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，拓扑空间上的侧度，Kolmogorov定理，Daniell积分，侧度的弱收敛，Skorohod表示，Prohorov定理，侧度空间上的弱拓扑，Lebesgue-Rohlin空间，Haar侧度，条件侧度与条件期望，遍历理论等。和第一卷一样，每章最后都附有非常丰富的补充和练习，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，ProhHorov空间，Young侧度等。书的最后有详尽的参考文献及历史注记。北京大学陈天权教授评价对此书“这是一本很好的研究生教材和教学参考书”。本书可作为高等学校数学类专业本科高年级和研究生的教材或预习课程的材料，也可供相关科学工作者参考。
目录
Preface to Volume 2
Chapter 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 Comments
References
Author Index
Subject Index
Chapter 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 Comments
References
Author Index
Subject Index
- 名称
- 类型
- 大小
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