测度论

出版时间：1998-8  出版社：世界图书出版公司  作者：Paul R.Halmos  页数：304  
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内容概要

My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of reference for the more advanced mathematician.     I have tried to keep to a minimum the amount of new and unusual terminology and notation. In the few places where my nomenclature differs from that in the existing literature of measure theory, I was motivated by an attempt to harmonize with the usage of other parts of mathematics. There are, for instance, sound algebraic reasons for using the terms "lattice" and "ring" for certain classes of sets--reasons which are more cogent than the similarities that caused Hausdorff to use "ring" and "field."

书籍目录

Preface Acknowledgments SECTION   0 Prerequisites CHAPTER Ⅰ: SETS AND CLASSES   1 Set inclusion   2 Unions and intersections   3 Limits， complements， and differences   4 Rings and algebras   5 Generated rings and -rings   6 Monotone classes CHAPTER Ⅱ: MEASURES AND OUTER MEASUR. ES   7 Measure on rings   8 Measure on intervals   9 Properties of measures   10 Outer measures   11 Measurable sets CHAPTER Ⅲ: EXTENSION OF MEASURES   12 Properties of induced measures   13 Extension， completion， and approximation   14 Inner measures  15 Lebesgue measure  16 Non measurable cetsCHAPTER Ⅳ: MEASURABLE FUNCTIONS  17 Measure spaces  18 Measurable functions  19 Combinations of measurable functions  20 Sequences of measurable functions  21 Pointwise convergence  22 Convergence in measureCHAPTER Ⅴ: INTEGRATION  23 Integrable simple functions  24 Sequences of integrable simple functions  25 Integrable functions  26 Sequences of integrable functions  27 Properties of IntegralsCHAPTER Ⅵ: GENERAL SET FUNCTIONS  28 Signed measures  29 Hahn and Jordan decompostions  30 Absolute continuity  31 The Radon-Nikodym theorem  32 Derivatives of signed measuresCHAPTER Ⅶ: PRODUCT SPACESCHAPTER Ⅷ: TRANSFORMATIONS AND FUNCTIONSCHAPTER Ⅸ: PROBABILITYCHAPTER Ⅹ: LOCALLY COMPACT SPACESCHAPTER Ⅺ: AHHR MEASURECHAPTER Ⅻ: MEASURE AND TOPOLOGY IN GROUPSReferencesBibliographyList of frequently used symbolsIndex