Probability based on Radon measures:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Chichester [u.a.]
Wiley
1980
|
| Schriftenreihe: | Wiley Series in probability and mathematical statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 232 S. Illustrationen |
| ISBN: | 0471278246 |
Internformat
MARC
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
PREFACE ix
PART I: MEASURE THEORY
CHAPTER 1: COMPACT, LOCALLY COMPACT, AND
COMPLETELY REGULAR SPACES 3
1.1. Compact spaces and nets 3
1.2. Functions on a compact space 4
1.3. Dint s theorem 5
1.4. Functions on a product of two compact spaces 5
1.5. Tensor products 6
1.6. Functions on an infinite product of compact spaces 6
1.7. Locally compact spaces 7
1.8. Functions on a locally compact space 9
1.9. Functions on a product of two locally compact spaces 10
1.10. Sigma-compactness 11
1.11. Completely regular spaces 12
1.12. Exercises 13
• 1.13. Remarks and references 15
CHAPTER 2: MEASURES ON LOCALLY COMPACT SPACES 16
2.1. Definitions, basic properties, and examples 16
2.2. Continuity under monotone convergence and the support of
a measure 17
2.3. The vague topology 19
2.4. Densities, transformed measures, and restrictions 20
2.5. Mixtures and decompositions 21
2.6. Defining a measure by its restrictions to an open covering 23
2.7. Product measures 24
2.8. Measures on an infinite product of compact spaces 26
2.9. Exercises 27
2.10. Remarks and references 30
v
vi CONTENTS
CHAPTER 3: THE LEBESGUE MEASURE AND
RELATED MEASURES 32
3.1. Transformation of the Lebesgue measure by a diffeomorphism 32
3.2. The Lebesgue measure on a Euclidean space 35
3.3. Manifolds 36
3.4. The geometric measure on a manifold 38
3.5. Decomposition of the Lebesgue measure 40
3.6. Exercises 43
3.7. Remarks and references 44
CHAPTER 4: INTEGRATION 46
4.1. The it-norm and the space of integrable functions 46
4.2. Integration 47
4.3. Integrability of extended real functions 48
4.4. Integrability of semicontinuous functions 51
4.5. The measure of a set 52
4.6. Convergence theorems 55
4.7. Null functions and null sets 57
4.8. Riesz-Fischer [s theorem and related results 58
4.9. Exercises 60
4.10. Remarks and references 64
CHAPTER 5: MEASURABILITY 66
5.1. Measurable mappings 66
5.2. Limits of measurable functions 68
5.3. Measurability and integrability 69
5.4. Measurable sets 70
5.5. Measurability and Borel sets 71
5.6. Exercises 72
5.7. Remarks and references 74
CHAPTER 6: LINEAR OPERATIONS ON MEASURES 76
6.1. Mixtures, transformed measures, and densities 76
6.2. Integration with respect to a mixture 77
6.3. Measurability and integrability with respect to a transformed
measure 80
6.4. Measurability and integrability with respect to a measure given by a
density 81
6.5. The Hilbert space L2(/x) 83
6.6. The space lf(ix) 84
CONTENTS vii
6.7. Radon-Nikodym s theorem 88
6.8. Exercises 89
6.9. Remarks and references 92
CHAPTER 7: BOUNDED MEASURES ON COMPLETELY
REGULAR SPACES 94
7.1. Introduction and the basic definition 94
7.2. Integration 96
7.3. Measurability and linear operations 100
7.4. Measures on infinite product spaces 103
7.5. Measures, linear functionals, and set functions 106
7.6. Exercises 110
7.7. Remarks and references 113
PART II: PROBABILITY
CHAPTER 8: PROBABILITY SPACES AND
RANDOM VARIABLES 117
8.1. Introduction and basic concepts 117
8.2. Convergence in probability 124
8.3. Sequences of independent random variables 129
8.4. Sums of independent random variables 133
8.5. Exercises 138
8.6. Remarks and references 141
CHAPTER 9: CONDITIONAL DISTRIBUTIONS 144
9.1. Introduction 144
9.2. Conditional expectations 144
9.3. The finite case 146
9.4. The conditional expectation operator 146
9.5. Extension to L(/x) 148
9.6. Conditional distributions: Introduction 149
9.7. Definition of a conditional distribution 151
9.8. The conditional distribution of a derived random variable 152
9.9. Continuity of the conditional distribution 152
9.10. The connection between conditional expectations and conditional
distributions 153
9.11. Decomposition of an underlying measure 156
9.12. Conditioning on Rn 157
9.13. Conditioning in a stochastic process 160
viii CONTENTS
9.14. The family of conditional distributions as a decomposition 162
9.15. Exercises 163
9.16. Remarks and references 166
CHAPTER 10: STOCHASTIC PROCESSES 169
10.1. Introduction: Increasing sample functions 169
10.2. Continuity in probability and almost surely 172
10.3. Separability 111
10.4. A sufficient condition for continuity of the sample function 179
10.5. Sample functions with simple discontinuities 183
10.6. A sufficient condition for simplicity of the sample function 187
10.7. Sample functions with left and right limits 192
10.8. Probability measures on function spaces 194
10.9. Exercises 196
10.10. Remarks and references 199
CHAPTER 11: RANDOM MEASURES AND POISSON
POINT DISTRIBUTIONS 204
11.1. Introduction and the consistency theorem for random
measures 204
11.2. Integration with respect to a random measure 206
11.3. Poisson random measures 208
11.4. Asymptotic behaviour of unnormalized empirical
distributions 214
11.5. Exercises 220
11.6. Remarks and references 224
REFERENCES 225
LIST OF SYMBOLS 228
SUBJECT INDEX 230
|
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| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:03:10Z |
| institution | BVB |
| isbn | 0471278246 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002318016 |
| oclc_num | 6487850 |
| open_access_boolean | |
| owner | DE-12 DE-384 DE-703 DE-355 DE-BY-UBR DE-824 DE-29T DE-19 DE-BY-UBM DE-20 DE-91G DE-BY-TUM DE-706 DE-83 DE-188 |
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| physical | XI, 232 S. Illustrationen |
| psigel | TUB-nveb |
| publishDate | 1980 |
| publishDateSearch | 1980 |
| publishDateSort | 1980 |
| publisher | Wiley |
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| series2 | Wiley Series in probability and mathematical statistics |
| spelling | Tjur, Tue 1945- Verfasser (DE-588)1201834333 aut Probability based on Radon measures Tue Tjur Chichester [u.a.] Wiley 1980 XI, 232 S. Illustrationen txt rdacontent n rdamedia nc rdacarrier Wiley Series in probability and mathematical statistics Locally compact spaces Probabilities Radon measures Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Radon-Maß (DE-588)4242168-8 gnd rswk-swf Radon-Maß (DE-588)4242168-8 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002318016&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tjur, Tue 1945- Probability based on Radon measures Locally compact spaces Probabilities Radon measures Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Radon-Maß (DE-588)4242168-8 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4242168-8 |
| title | Probability based on Radon measures |
| title_auth | Probability based on Radon measures |
| title_exact_search | Probability based on Radon measures |
| title_full | Probability based on Radon measures Tue Tjur |
| title_fullStr | Probability based on Radon measures Tue Tjur |
| title_full_unstemmed | Probability based on Radon measures Tue Tjur |
| title_short | Probability based on Radon measures |
| title_sort | probability based on radon measures |
| topic | Locally compact spaces Probabilities Radon measures Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Radon-Maß (DE-588)4242168-8 gnd |
| topic_facet | Locally compact spaces Probabilities Radon measures Wahrscheinlichkeitsrechnung Radon-Maß |
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