測(cè)度論

前言

為了更好地借鑒國外數(shù)學(xué)教育與研究的成功經(jīng)驗(yàn)，促進(jìn)我國數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，本著“為我國熱愛數(shù)學(xué)的青年創(chuàng)造一個(gè)較好的學(xué)習(xí)數(shù)學(xué)的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學(xué)叢書”。該叢書主要包含國外反映近代數(shù)學(xué)發(fā)展的純數(shù)學(xué)與應(yīng)用數(shù)學(xué)方面的優(yōu)秀書籍，天元基金邀請(qǐng)國內(nèi)各個(gè)方向的知名數(shù)學(xué)家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數(shù)學(xué)研究生教學(xué)的水平，暫把選書的目標(biāo)確定在研究生教材上。當(dāng)然，有的書也可作為高年級(jí)本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對(duì)本叢書的選題、印刷、銷售等工作提出批評(píng)和建議。

內(nèi)容概要

　　《測(cè)度論（第2卷）（影印版）》是作者在莫斯科國立大學(xué)數(shù)學(xué)力學(xué)系的講稿基礎(chǔ)上編寫而成的。第二卷介紹測(cè)度論的專題性的內(nèi)容，特別是與概率論和點(diǎn)集拓?fù)溆嘘P(guān)的課題：Borel集，Baire集，Souslin集，拓?fù)淇臻g上的測(cè)度，Kolmogorov定理，Daniell積分，測(cè)度的弱收斂，Skorohod表示，Prohorov定理，測(cè)度空間上的弱拓?fù)洌琇ebesgue-Rohlin空間，Haar測(cè)度，條件測(cè)度與條件期望，遍歷理論等。每章最后都附有非常豐富的補(bǔ)充與練習(xí)，其中包含許多有用的知識(shí)，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(cè)度等。書的最后有詳盡的參考文獻(xiàn)及歷史注記。這是一本很好的研究生教材和教學(xué)參考書。

作者簡介

作者：（俄羅斯）博根切維（V.I.Bogachev）

書籍目錄

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 Lusin's 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

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《測(cè)度論(第2卷)(影印版)》：天元基金影印數(shù)學(xué)叢書

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