《測度論(第2卷)(影印版)》是作者在莫斯科國立大學數(shù)學力學系的講稿基礎上編寫而成的。第二卷介紹測度論的專題性的內(nèi)容，特別是與概率論和點集拓撲有關的課題：Borel集，Baire集，Souslin集，拓撲空間上的測度，Kolmogorov定理，Daniell積分，測度的弱收斂，Skorohod表示，Prohorov定理，測度空間上的弱拓撲，Lebesgue-Rohlin空間，Haar測度，條件測度與條件期望，遍歷理論等。每章最后都附有非常豐富的補充與練習，其中包含許多有用的知識，例如：Skorohod空間，Blackwell空間，Marik空間，Radon空間，推廣的Lusin定理，容量，Choquet表示，Prohorov空間，Young測度等。書的最后有詳盡的參考文獻及歷史注記。這是一本很好的研究生教材和教學參考書。
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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 conditionalexpectations 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
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 conditionalexpectations 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
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