《Measure Theory》求取 ⇩

作者 Paul R.Halmos 编者
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Measure Theory PDF电子版封面 Paul R.Halmos

O．Prerequisites1

CHAPTER Ⅰ：SETS AND CLASSES9

1．Set inclusion9

2．Unions and intersections11

3．Limits，complements，and differences16

4．Rings and algebras19

5．Generated rings and σ-rings22

6．Monotone classes26

CHAPTER Ⅱ：MEASURES AND OUTER MEASURES30

7．Measure on rings30

8．Measure or intervals32

9．Properties of measures37

10．Outer measures41

11．Measurable Sets44

CHAPTER Ⅲ：EXTENSION OF MEASURES49

12．Properties of induced measures49

13．Extension，completion，and approximation54

14．Inner measures58

15．Lebesgue measure62

16．Non measurable sets67

CHAPTER Ⅳ：MEASURABLE FUNCTIONS73

17．Measure spaces73

18．Measurable functions76

19．Combinations of measurable functions80

20．Sequences of measurable functions84

21．Pointwise convergence86

22．Convergence in measure90

CHAPTER Ⅴ：INTEGRATION95

23．Integrable simple functions95

24．Sequences of integrable simple functions98

25．Integrable functions102

26．Sequences of integrable functions107

27．Properties of integrals112

CHAPTER Ⅵ：GENERAL SET FUNCTIONS117

28．Signed measures117

29．Hahn and Jordan decompositions120

30．Absolute continuity124

31．The Radon-Nikodym theorem128

32．Derivatives of signed measures132

CHAPTER Ⅶ：PRODUCT SPACES137

33．Cartesian products137

34．Sections141

35．Product measures143

36．Fubini＇s theorem145

37．Finite dimensional product spaces150

38．Infinite dimensional product spaces154

CHAPTER Ⅷ：TRANSFORMATIONS AND FUNCTIONS161

39．Measurable transformations161

40．Measure rings165

41．The isomorphism theorem171

42．Function spaces174

43．Set functions and point functions178

CHAPTER Ⅸ：PROBABILITY184

44．Heuristic introduction184

45．Independence151

46．Series of independent functions196

47．The law of large numbers201

48．Conditional probabilities and expectations206

49．Measures on product spaces211

CHAPTER Ⅹ：LOCALLY COMPACT SPACES216

50．Topological lemmas216

51．Borel sets and Baire sets219

52．Regular measures223

53．Generation of Borel measures231

54．Regular contents237

55．Classes of continuous functions240

56．Linear functionals243

CHAPTER Ⅺ：HAAR MEASURE250

57．Full subgroups250

58．Existence251

59．Measurable groups257

60．Uniqueness262

CHAPTER Ⅻ：MEASURE AND TOPOLOGY IN GROUPS266

61．Topology in terms of measure266

62．Weil topology270

63．Quotient groups277

64．The regularity of Haar measure282

References291

Bibliography293

List of frequently used symbols297

Index299

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