Measures, integrals and martingales:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2017
|
| Ausgabe: | Second edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xvii, 476 Seiten Illustrationen 25 cm |
| ISBN: | 9781316620243 |
Internformat
MARC
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| 100 | 1 | |a Schilling, René L. |d 1969- |0 (DE-588)122050045 |4 aut | |
| 245 | 1 | 0 | |a Measures, integrals and martingales |c René L. Schilling |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2017 | |
| 300 | |a xvii, 476 Seiten |b Illustrationen |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Measure theory | |
| 650 | 4 | |a Integrals | |
| 650 | 4 | |a Martingales (Mathematics) | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
List of Symbols page x
Prelude xiii
Dependence Chart xvii
1 Prologue 1
Problems 5
2 The Pleasures of Counting 6
Problems 14
3 er-Algebras 16
Problems 21
4 Measures 23
Problems 28
5 Uniqueness of Measures 32
Problems 37
6 Existence of Measures 39
Existence of Lebesgue measure in R 46
Existence of Lebesgue measure in R” 47
Problems 50
7 Measurable Mappings 53
Problems 58
8 Measurable Functions 60
Problems 69
9 Integration of Positive Functions 72
Problems 79
V
vi Contents
10 Integrals of Measurable Functions 82
Problems 87
11 Null Sets and the ‘Almost Everywhere’ 89
Problems 93
12 Convergence Theorems and Their Applications 96
Application 1: Parameter-Dependent Integrals 99
Application 2: Riemann vs. Lebesgue Integration 101
Improper Riemann Integrals 105
Examples 107
Problems 110
13 The Function Spaces CP 116
Convergence in Cp and Completeness 120
Convexity and Jensen’s Inequality 125
Convexity Inequalities in K+ 128
Problems 132
14 Product Measures and Fubini’s Theorem 136
Integration by Parts and Two Interesting Integrals 143
Distribution Functions 146
Minkowski’s Inequality for Integrals 148
More on Measurable Functions 149
Problems 149
15 Integrals with Respect to Image Measures 154
Convolutions 157
Regularization 160
Problems 162
16 Jacobi’s Transformation Theorem 164
A useful Generalization of the Transformation Theorem 170
Images of Borel Sets 172
Polar Coordinates and the Volume of the Unit Ball 176
Surface Measure on the Sphere 181
Problems 183
17 Dense and Determining Sets 186
Dense Sets 186
Determining Sets 191
Problems 194
Contents vii
18 Hausdorff Measure 197
Constructing (Outer) Measures 197
Hausdorff Measures 202
Hausdorff Dimension 209
Problems 212
19 The Fourier Transform 214
Injectivity and Existence of the Inverse Transform 217
The Convolution Theorem 220
The Riemann—Lebesgue Lemma 221
The Wiener Algebra, Weak Convergence and Plancherel 223
The Fourier Transform in S(M”) 227
Problems 228
20 The Radon-Nikod^m Theorem 230
Problems 236
21 Riesz Representation Theorems 238
Bounded and Positive Linear Functionals 238
Duality of the Spaces 1 Հբ oo 241
The Riesz Representation Theorem for Cc(X) 243
Vague and Weak Convergence of Measures 249
Problems 255
22 Uniform Integrability and Vitali’s Convergence Theorem 258
Different Forms of Uniform Integrability 264
Problems 272
23 Martingales 275
Problems 286
24 Martingale Convergence Theorems 288
Problems 298
25 Martingales in Action 300
The Radon—Nikodym Theorem 300
Martingale Inequalities 308
The Hardy—Littlewood Maximal Theorem 310
Lebesgue’s Differentiation Theorem 314
The Calderon—Zygmund Lemma 318
Problems 319
viii Contents
26 Abstract Hilbert Spaces 322
Convergence and Completeness 327
Problems 338
27 Conditional Expectations 341
Extension from L2 to LP 345
Monotone Extensions 347
Properties of Conditional Expectations 349
Conditional Expectations and Martingales 355
On the Structure of Subspaces of L2 357
Separability Criteria for the Spaces If(X, ¿Z, /1) 361
Problems 366
28 Orthonormal Systems and Their Convergence Behaviour 370
Orthogonal Polynomials 370
The Trigonometric System and Fourier Series 376
The Haar System 383
The Haar Wavelet 389
The Rademacher Functions 393
Well-Behaved Orthonormal Systems 396
Problems 407
Appendix A lim inf and lim sup 409
Appendix B Some Facts from Topology 415
Continuity in Euclidean Spaces 415
Metric Spaces 416
Appendix C The Volume of a Parallelepiped 421
Appendix D The Integral of Complex-Valued Functions 423
Appendix E Measurability of the Continuity Points of a Function 425
Appendix F Vitali’s Covering Theorem 427
Appendix G Non-measurable Sets 429
Appendix H Regularity of Measures 437
Appendix IA Summary of the Riemann Integral 441
The (Proper) Riemann Integral 441
The Fundamental Theorem of Integral Calculus 450
Integrals and Limits 455
Improper Riemann Integrals 458
References 465
Index 469
|
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| discipline | Mathematik |
| edition | Second edition |
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| isbn | 9781316620243 |
| language | English |
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| spelling | Schilling, René L. 1969- (DE-588)122050045 aut Measures, integrals and martingales René L. Schilling Second edition Cambridge Cambridge University Press 2017 xvii, 476 Seiten Illustrationen 25 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Measure theory Integrals Martingales (Mathematics) Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Maßtheorie (DE-588)4074626-4 s Integrationstheorie (DE-588)4138369-2 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029754479&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schilling, René L. 1969- Measures, integrals and martingales Measure theory Integrals Martingales (Mathematics) Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4138369-2 (DE-588)4074626-4 (DE-588)4151278-9 |
| title | Measures, integrals and martingales |
| title_auth | Measures, integrals and martingales |
| title_exact_search | Measures, integrals and martingales |
| title_full | Measures, integrals and martingales René L. Schilling |
| title_fullStr | Measures, integrals and martingales René L. Schilling |
| title_full_unstemmed | Measures, integrals and martingales René L. Schilling |
| title_short | Measures, integrals and martingales |
| title_sort | measures integrals and martingales |
| topic | Measure theory Integrals Martingales (Mathematics) Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Measure theory Integrals Martingales (Mathematics) Integrationstheorie Maßtheorie Einführung |
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