章节目录
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 cets CHAPTER 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 measure CHAPTER Ⅴ: INTEGRATION 23 Integrable simple functions 24 Sequences of integrable simple functions 25 Integrable functions 26 Sequences of integrable functions 27 Properties of Integrals CHAPTER Ⅵ: GENERAL SET FUNCTIONS 28 Signed measures 29 Hahn and Jordan decompostions 30 Absolute continuity 31 The Radon-Nikodym theorem 32 Derivatives of signed measures CHAPTER Ⅶ: PRODUCT SPACES CHAPTER Ⅷ: TRANSFORMATIONS AND FUNCTIONS CHAPTER Ⅸ: PROBABILITY CHAPTER Ⅹ: LOCALLY COMPACT SPACES CHAPTER Ⅺ: AHHR MEASURE CHAPTER Ⅻ: MEASURE AND TOPOLOGY IN GROUPS …… References Bibliography List of frequently used symbols Index
内容简介
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."
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