Measure TheoryPDF电子书下载

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O．Prerequisites 1

CHAPTER Ⅰ： SETS AND CLASSES 9

1．Set inclusion 9

2．Unions and intersections 11

3．Limits，complements，and differences 16

4．Rings and algebras 19

5．Generated rings and σ-rings 22

6．Monotone classes 26

CHAPTER Ⅱ： MEASURES AND OUTER MEASURES 30

7．Measure on rings 30

8．Measure or intervals 32

9．Properties of measures 37

10．Outer measures 41

11．Measurable Sets 44

CHAPTER Ⅲ： EXTENSION OF MEASURES 49

12．Properties of induced measures 49

13．Extension，completion，and approximation 54

14．Inner measures 58

15．Lebesgue measure 62

16．Non measurable sets 67

CHAPTER Ⅳ： MEASURABLE FUNCTIONS 73

17．Measure spaces 73

18．Measurable functions 76

19．Combinations of measurable functions 80

20．Sequences of measurable functions 84

21．Pointwise convergence 86

22．Convergence in measure 90

CHAPTER Ⅴ： INTEGRATION 95

23．Integrable simple functions 95

24．Sequences of integrable simple functions 98

25．Integrable functions 102

26．Sequences of integrable functions 107

27．Properties of integrals 112

CHAPTER Ⅵ： GENERAL SET FUNCTIONS 117

28．Signed measures 117

29．Hahn and Jordan decompositions 120

30．Absolute continuity 124

31．The Radon-Nikodym theorem 128

32．Derivatives of signed measures 132

CHAPTER Ⅶ： PRODUCT SPACES 137

33．Cartesian products 137

34．Sections 141

35．Product measures 143

36．Fubini＇s theorem 145

37．Finite dimensional product spaces 150

38．Infinite dimensional product spaces 154

CHAPTER Ⅷ： TRANSFORMATIONS AND FUNCTIONS 161

39．Measurable transformations 161

40．Measure rings 165

41．The isomorphism theorem 171

42．Function spaces 174

43．Set functions and point functions 178

CHAPTER Ⅸ： PROBABILITY 184

44．Heuristic introduction 184

45．Independence 151

46．Series of independent functions 196

47．The law of large numbers 201

48．Conditional probabilities and expectations 206

49．Measures on product spaces 211

CHAPTER Ⅹ： LOCALLY COMPACT SPACES 216

50．Topological lemmas 216

51．Borel sets and Baire sets 219

52．Regular measures 223

53．Generation of Borel measures 231

54．Regular contents 237

55．Classes of continuous functions 240

56．Linear functionals 243

CHAPTER Ⅺ： HAAR MEASURE 250

57．Full subgroups 250

58．Existence 251

59．Measurable groups 257

60．Uniqueness 262

CHAPTER Ⅻ： MEASURE AND TOPOLOGY IN GROUPS 266

61．Topology in terms of measure 266

62．Weil topology 270

63．Quotient groups 277

64．The regularity of Haar measure 282

References 291

Bibliography 293

List of frequently used symbols 297

Index 299

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