测度论

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 σ-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

Souslin sets as projections

K-analytic and F-analytic sets

Blackwell spaces

Mappings of Souslin spaces

Measurability in normed spaces

The Skorohod space

Exercises

Chapter 7 Measures on topological spaces

7.1.Borel， Baire and Radon measures

7.2.τ-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

Measurability on products

Marik spaces

Separable measures

Diffused and atomless measures

Completion regular measures

Radon spaces

Supports of measures

Generalizations of Lusins theorem

Metric outer measures

Capacities

Covariance operators and means of measures

The Choquet representation

Convolution

Measurable linear functions

Convex measures

Pointwise convergence

Infinite Radon measures

Exercises

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

Prohorov spaces

The weak sequential completeness of spaces of measures

The A-topology

Continuous mappings of spaces of measures

The separability of spaces of measures

Young measures

Metrics on spaces of measures

Uniformly distributed sequences

Setwise convergence of measures

Stable convergence and ws-topology

Exercises

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

Extremal preimages of measures and uniqueness

Existence of atomlees measures

Invariant and quasi-invariant measures of transformations

Point and Boolean isomorphisms

Almost homeomorphisms Measures with given marginal projections

The Stonerepresentation

The Lyapunov theorem

Exercises

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

Disintegrations

Strong liftings

Zero-one laws

Laws of large numbers

Gibbs measures

Triangular mappings

Exercises

Bibliographical and Historical Comments

References

Author Index

Subject Index

《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。