Measure TheorySpringer, 19 déc. 2013 - 304 pages 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 |
Table des matières
| 2 | |
| 3 | |
| 9 | |
Unions and intersections | 15 |
Limits complements and differences | 16 |
Rings and algebras | 21 |
Generated rings and orings 6 Monotone classes | 27 |
MEASURES AND OUTER MEASURES 7 Measure on rings | 30 |
32 | 132 |
PRODUCT SPACES | 137 |
Product measures | 143 |
37 | 150 |
Infinite dimensional product spaces | 158 |
Measurable transformations | 164 |
41 | 171 |
Set functions and point functions | 178 |
Measure on intervals | 32 |
Properties of measures | 37 |
Outer measures | 42 |
Measurable sets | 46 |
EXTENSION OF MEASURES 12 Properties of induced measures | 49 |
Extension completion and approximation | 54 |
Inner measures | 58 |
Lebesgue measure | 62 |
Non measurable sets | 72 |
9 | 79 |
11 | 80 |
19 | 83 |
Pointwise convergence | 86 |
22 | 90 |
Convergence in measure | 92 |
Sequences of integrable simple functions | 101 |
26 | 107 |
GENERAL SET FUNCTIONS | 117 |
30 | 124 |
Absolute continuity | 125 |
The RadonNikodym theorem | 131 |
44 | 184 |
Independence | 192 |
SECTION PAGE | 201 |
49 | 211 |
Borel sets and Baire sets | 219 |
Generation of Borel measures | 231 |
54 | 237 |
Linear functionals | 243 |
HAAR MEASURE | 250 |
58 | 251 |
Measurable groups | 257 |
MEASURE AND TOPOLOGY IN GROUPS | 266 |
62 | 270 |
Quotient groups | 277 |
67 | 291 |
Measure spaces | 294 |
Measurable functions | 296 |
| 297 | |
| 299 | |
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Expressions et termes fréquents
A₁ absolutely continuous algebra assertion Baire set Borel measure Borel set Cartesian product class of sets compact set converges in measure countable decreasing sequence defined denote E₁ element empty exists a set extended real valued F₁ F₂ follows from Theorem function ƒ ƒ and g Haar measure Hausdorff space hence Hint implies inner regular integrable function intersection Lebesgue measure measurable function measurable group measurable set measurable subset measurable transformation measure induced measure ring measure space measure zero measure µ metric space monotone negative o-finite o-ring open set outer measure outer regular positive integer positive measure positive number product space Proof prove real line real numbers respect sequence f sequence of sets set function set of positive sets of finite signed measure subgroup Theorem F tion topological group unique write µ(En