Useful as a text for students and a reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory most useful for its application in modern analysis. Coverage includes sets and classes, measures and outer measures, Haar measure and measure and topology in groups. From the reviews: "Will serve the interested student to find his way to active and creative work in the field of Hilbert space theory." --MATHEMATICAL REVIEWS
Author(s): Paul R. Halmos
Series: Graduate Texts in Mathematics, 18
Publisher: Springer
Year: 1974
Language: English
Pages: 316
Tags: Математика;Функциональный анализ;Теория меры;
PREFACE......Page 5
ACKNOWLEDGMENTS......Page 7
CONTENTS......Page 9
0. Prerequisites......Page 13
1. Set inclusion......Page 21
2. Unions and intersections......Page 23
3. Limits, complements, and differences......Page 28
4. Rings and algebras......Page 31
5. Generated rings and $sigma$-rings......Page 34
6. Monotone classes......Page 38
7. Measure on rings......Page 42
8. Measure on intervals......Page 44
9. Properties of measures......Page 49
10. Outer measures......Page 53
11. Measurable sets......Page 56
12. Properties of induced measures......Page 61
13. Extension, completion, and approximation......Page 66
14. Inner measures......Page 70
15. Lebesgue measure......Page 74
16. Non measurable sets......Page 79
17. Measure spaces......Page 85
18. Measurable functions......Page 88
19. Combinations of measurable functions......Page 92
20. Sequences of measurable functions......Page 96
21. Pointwise convergence......Page 98
22. Convergence in measure......Page 102
23. Integrable simple functions......Page 107
24. Sequences of integrable simple functions......Page 110
25. Integrable functions......Page 114
26. Sequences of integrable functions......Page 119
27. Properties of integrals......Page 124
28. Signed measures......Page 129
29. Hahn and Jordan decompositions......Page 132
30. Absolute continuity......Page 136
31. The Radon-Nikodym theorem......Page 140
32. Derivatives of signed measures......Page 144
33. Cartesian products......Page 149
34. Sections......Page 153
35. Product measures......Page 155
36. Fubini's theorem......Page 157
37. Finite dimensional product spaces......Page 162
38. Infinite dimensional product spaces......Page 166
39. Measurable transformations......Page 173
40. Measure rings......Page 177
41. The isomorphism theorem......Page 183
42. Function spaces......Page 186
43. Set functions and point functions......Page 190
44. Heuristic introduction......Page 196
45. Independence......Page 203
46. Series of independent functions......Page 208
47. The law of large numbers......Page 213
48. Conditional probabilities and expectations......Page 218
49. Measures on product spaces......Page 223
50. Topological lemmas......Page 228
51. Borel sets and Baire sets......Page 231
52. Regular measures......Page 235
53. Generation of Borel measures......Page 243
54. Regular contents......Page 249
55. Classes of continuous functions......Page 252
56. Linear functional......Page 255
57. Full subgroups......Page 262
58. Existence......Page 263
59. Measurable groups......Page 269
60. Uniqueness......Page 274
61. Topology in terms of measure......Page 278
62. Weil topology......Page 282
63. Quotient groups......Page 289
64. The regularity of Haar measure......Page 294
References......Page 303
Bibliography......Page 305
List of frequently used symbols......Page 309
Index......Page 311
