测度论

作者简介

《测度论(英文版)》讲述了：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.

书籍目录

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

前言

　　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.

内容概要

　　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.

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