测度论
2009-6
世界图书出版公司
杜波
210
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In presenting this treatment of homological algebra， it is a pleasureto acknowledge the help and encouragement which I have had fromall sides. Homological algebra arose from many sources in algebra andtopology. Decisive examples came from the study of group extensionsand their factor sets， a subject I learned in joint work with OTTO SCHIL-LING. A further development of homological ideas， with a view to theirtopological applications， came in my long collaboration with SAHUELEZLENBERG; to both collaborators， especial thanks. For many yearsthe Air Force Office of Scientific Research supported my researchprojects on various subjects now summarized here; it is a pleasure toacknowledge their lively understanding Of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented onmy entire manuscript; their advice has led to many improvements.ANDERS KOCK and JACOUES RIGUET have read the entire galley proofand caught many slips and obscurities. Among the others whose sug-gestions have served me well， I note FRANK ADAMS， LOUIS AUSLANDER，WILFRED COCKCROFT， ALBRECHT DOLD， GEOFFREY HORROCKS， FRIED-RICH KASCH， JOHANN LEICHT， ARUNAS LIULEVIC1US， JOHN MOORE， DIE-TBR PUFFE， JOSEPH YAO， and a number of my current students at theUniversity of Chicago —— not to mention the auditors of my lecturesat Chicago， Heidelberg， Bonn， Frankfurt， and Aarhus. My wife， DonoTHY，has cheerfully typed more versions of more chapters than she wouldlike to count. Messrs. SPRINTER have been unfailingly courteous in thepreparation of the book; in particular， I am grateful to F. K. SCHMIDT，the Editor of this series， for his support. To all these and others whohave helped me， I express my best thanks.
本书是Springer研究生数学丛书之一，对测度论的讲述完全不同于一般的教程。该书将概率论作为测度论的必不缺少的一部分，所以书中的许多例子都是来自概率论，如独立性、马尔科夫过程、条件期望这些都作为本书的组成部分而不是将其置于附录中作为补充。特别是对Sigma代数做了较多的研究，而不是拿来即用。运用伪度量而不是度量给出了集合空间和函数空间的距离更直观的定义。
Saunders Mac Lane was born on August 4，1909in Connecticut. He studied at Yale Universityand then at the University of Chicago and atG6ttingen， where he received the D. Phil. in 1934.He has taught at Harvard， Cornel1 and theUniversity of Chicago.Mac Lanes initial research was in logic andin algebraic number theory （valuation theory）.With Samuel Eilenberg he published fifteenpapers on algebraic topology. A number of theminvolved the initial steps in the cohomologyof groups and in other aspects of homologicalalgebra - as well as the discovery of categorytheory. His famous undergraduate textbookSurvey of modern algebra， written jointly withG. Birkhoff， has remained in print for over50 years. Mac Lane is also the author of severalother highly successful books.
IntroductionO. Conventions and Notation 1.Notation: Euclidean space 2.Operations involving 3.Inequalities and inclusions 4.A space and its subsets 5.Notation: generation of classes of sets 6.Product sets 7.Dot notation for an index set 8.Notation: sets defined by conditions on functions 9.Notation: open and closed sets 10.Limit of a function at a point 11.Metric spaces 12.Standard metric space theorems 13.Pseudometric spacesI. Operations on Sets I.Unions and intersections 2.The symmetric difference operator 3.Limit operations on set sequences 4.Probabilistic interpretation of sets and operations on themII. Classes of Subsets of a Space 1.Set algebras 2.Examples 3.The generation of set algebras 4.The Borel sets of a metric space 5.Products of set algebras 6.Monotone classes of setsIII. Set Functions 1.Set function definitions 2.Extension of a finitely additive set function 3.Products of set functions 4.Heuristics on a algebras and integration 5.Measures and integrals on a countable space 6.Independence and conditional probability (preliminary discussion) 7.Dependence examples 8.Inferior and superior limits of sequences of measurable sets 9.Mathematical counterparts of coin tossing 10.Setwise convergence of measure sequences 11.Outer measure 12.Outer measures of countable subsets of R 13.Distance on a set algebra defined by a subadditive set function 14.The pseudometric space defined by an outer measure 15.Nonadditive set functionsIV. Measure Spaces 1.Completion of a measure space (S,$,~,) 2.Generalization of length on R 3.A general extension problem 4.Extension of a measure defined on a set algebra 5.Application to Borel measures 6.Strengthening of Theorem 5 when the metric space S is complete and separable 7.Continuity properties of monotone functions 8.The correspondence between monotone increasing functions on R and measures on B(R) 9.Discrete and continuous distributions on R 10.Lebesgue-Stieltjes measures on R/v and their corresponding monotone functions 11.Product measures 12.Examples of measures on RN 13.Marginal measures 14.Coin tossing 15.The Caratheodory measurability criterion 16.Measure hulls V. Measurable Functions 1.Function measurability 2.Function measurability properties 3.Measurability and sequential convergence 4.Baire functions 5.Joint distributions 6.Measures on function (coordinate) space……VI. IntezrationVII. Hilbert SoaceVIII. Convergence of Measure SequencesIX. Signed MeasuresX. Measures and Functions of Bounded Variation on RXI. Conditional Expectations; Martingale TheoryNotationIndex
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