简介
《测度论》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."
目录
Preface AcknowledgmentsO．Prerequisites CHAPTER I：SETS AND CLASSES1. Set inclusion 2. Unions and intersections 3. Limits，complements，and differences4. Ringsg and algebras5. Generated rings and rings6. Monotone classes CHAPTER II:MEASURES AND OUTER MEASURES 7. Measure on rings 8. Measure on intervals 9. Properties of measures 10. Outer measures 11. Measurable sets CHAPTER III：EXTENSION OF MEASURES12．Properties of induced measures 13．Extension，completion，and approximation 14. Inner measures 15. Lebesgue measure 16．Non measurable setsn CHAPTER IV：MEASURABLE FUNCTIONS17．Measure space 18．Measurable functions19．Combinations of measurable functions20．Sequences of measurable functions 21．Pointwise convergence22．Convergenceinmeasure CHAPTER V：INTEGRATI0N23. Integrable simple functions 24. Sequences of integrable simple functions25. Integrable functions 26. Sequences of integrable functions 27. Properdes ofintegrals CHAPTER VI：GENERAL SET FUNCTIONS28．Signed measures 29．Hahn and Jordan decompositions30．Absolute continuity 31．The Radon-Nikodym theorem 32．Derivatives of signed measures CHAPTERVII：PRODUCT SPACES33. Cartesian products 34. Sections 35. Product measures 36. Fubini’s theorem 37. Finite dimensional product spaces 38. Infinite dimensional product spacesCHAPTER VIII：TRANSFORMATIONS AND FUNCTIONS39． Measurable transformations40． Measure rings 41． The isomorphism theorem 42． Function spaces 43． Set functions and point functions CHAPTER IX：PROBABILITY 44． Heuristic introduction 45． Independence46． Series ofindependent functions 47． The law oflarge numbers 48． Conditional probabilities and expectations49． Measures on product spaces CHAPTER X：LOCALLY COMPACT SPACES50． Topological 1emmas 51． Borel sets and Baire sets 52． Regular measures 53． Generation of Borel measures 54． Regular contents 55． Classes of continuous functions 56． Linear functionals CHAPTER XI：HAAR MEASURE57． Full subgroups 58． Existence 59． Measurable groups 60． Uniqueness CHAPTER XII：MEASURE AND TOPOLOGY IN GROUPS61． Topology in terms of measure 62． Well topology 63． Quotient groups 64． The regularity of Haar measure References Bibliography List of frequently used symbolsn Index
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