Measure, integral and probability:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London ; Berlin [u.a.]
Springer
2002
|
| Edition: | 4. print. |
| Series: | Springer undergraduate mathematics series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 227 S. graph. Darst. |
| ISBN: | 3540762604 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804144900631953408 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
1. MOTIVATION AND PRELIMINARIES 1
1.1 NOTATION AND BASIC SET THEORY 2
1.1.1 SETS AND FUNCTIONS 2
1.1.2 COUNTABLE AND UNCOUNTABLE SETS IN E 4
1.1.3 TOPOLOGICAL PROPERTIES OF SETS IN E 5
1.2 THE RIEMANN INTEGRAL: SCOPE AND LIMITATIONS 7
1.3 CHOOSING NUMBERS AT RANDOM 12
2. MEASURE 15
2.1 NULL SETS 15
2.2 OUTER MEASURE 20
2.3 LEBESGUE MEASURABLE SETS AND LEBESGUE MEASURE 27 2.4 BASIC
PROPERTIES OF LEBESGUE MEASURE 35
2.5 BOREL SETS 40
2.6 PROBABILITY 45
.2.6.1 PROBABILITY SPACE 46
2.6.2 EVENTS: CONDITIONING AND INDEPENDENCE 46
2.7 PROOFS OF PROPOSITIONS 49
3. MEASURABLE FUNCTIONS 53
3.1 THE EXTENDED REAL LINE 53
3.2 DEFINITION 53
3.3 EXAMPLES 57
3.4 PROPERTIES 58
3.5 PROBABILITY 64
3.5.1 RANDOM VARIABLES 64
IX
IMAGE 2
X CONTENTS
3.5.2 SIGMA FIELDS GENERATED BY RANDOM VARIABLES 65
3.5.3 PROBABILITY DISTRIBUTIONS 66
3.5.4 INDEPENDENCE OF RANDOM VARIABLES 68
3.6 PROOFS OF PROPOSITIONS 69
4. INTEGRAL 71
4.1 DEFINITION OF THE INTEGRAL 71
4.2 MONOTONE CONVERGENCE THEOREMS 78
4.3 INTEGRABLE FUNCTIONS 82
4.4 THE DOMINATED CONVERGENCE THEOREM 87
4.5 RELATION TO THE RIEMANN INTEGRAL 92
4.6 APPROXIMATION OF MEASURABLE FUNCTIONS 97
4.7 PROBABILITY 100
4.7.1 INTEGRATION WITH RESPECT TO PROBABILITY DISTRIBUTIONS . . .. 100
4.7.2 ABSOLUTELY CONTINUOUS MEASURES: EXAMPLES OF DENSITIES . . 102
4.7.3 EXPECTATION OF A RANDOM VARIABLE 109
4.7.4 CHARACTERISTIC FUNCTION 110
4.8 PROOFS OF PROPOSITIONS I LL
5. SPACES OF INTEGRABLE FUNCTIONS 115
5.1 THE SPACE L 1 116
5.2 THE HILBERT SPACE L 2 121
5.2.1 PROPERTIES OF THE L 2 -NORM 122
5.2.2 INNER PRODUCT SPACES 124
5.2.3 ORTHOGONALITY 127
5.3 THE L P SPACES: COMPLETENESS 128
5.4 PROBABILITY 134
5.4.1 MOMENTS 134
5.4.2 INDEPENDENCE 138
5.5 PROOFS OF PROPOSITIONS 141
6. PRODUCT MEASURES 145
6.1 MULTI-DIMENSIONAL LEBESGUE MEASURE 145
6.2 PRODUCT R-FIELDS 146
6.3 CONSTRUCTION OF THE PRODUCT MEASURE 148
6.4 FUBINI S THEOREM 155
6.5 PROBABILITY 159
6.5.1 JOINT DISTRIBUTIONS 159
6.5.2 INDEPENDENCE AGAIN 161
6.5.3 CONDITIONAL PROBABILITY 163
6.5.4 CHARACTERISTIC FUNCTIONS DETERMINE DISTRIBUTIONS 165 6.6 PROOFS OF
PROPOSITIONS 168
IMAGE 3
CONTENTS XI
7. LIMIT THEOREMS 171
7.1 MODES OF CONVERGENCE 171
7.2 PROBABILITY 173
7.2.1 CONVERGENCE IN PROBABILITY 175
7.2.2 WEAK LAW OF LARGE NUMBERS 179
7.2.3 BOREL-CANTELLI LEMMAS 184
7.2.4 STRONG LAW OF LARGE NUMBERS 188
7.2.5 WEAK CONVERGENCE 196
7.2.6 CENTRAL LIMIT THEOREM 201
7.3 PROOFS OF PROPOSITIONS 208
8. SOLUTIONS TO EXERCISES 209
9. APPENDIX 219
REFERENCES 223
INDEX 225
IMAGE 4
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T23:10:54Z |
| institution | BVB |
| isbn | 3540762604 |
| language | English |
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| physical | XI, 227 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
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| publisher | Springer |
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| series2 | Springer undergraduate mathematics series |
| spelling | Capiński, Marek 1951- Verfasser (DE-588)172897866 aut Measure, integral and probability Marek Capiński and Ekkehard Kopp 4. print. London ; Berlin [u.a.] Springer 2002 XI, 227 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer undergraduate mathematics series Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Lebesgue-Integral (DE-588)4034949-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 s DE-604 Lebesgue-Integral (DE-588)4034949-4 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Kopp, Peter E. 1944- Verfasser (DE-588)120339889 aut SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021962335&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Capiński, Marek 1951- Kopp, Peter E. 1944- Measure, integral and probability Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Lebesgue-Integral (DE-588)4034949-4 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4034949-4 (DE-588)4074626-4 |
| title | Measure, integral and probability |
| title_auth | Measure, integral and probability |
| title_exact_search | Measure, integral and probability |
| title_full | Measure, integral and probability Marek Capiński and Ekkehard Kopp |
| title_fullStr | Measure, integral and probability Marek Capiński and Ekkehard Kopp |
| title_full_unstemmed | Measure, integral and probability Marek Capiński and Ekkehard Kopp |
| title_short | Measure, integral and probability |
| title_sort | measure integral and probability |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Lebesgue-Integral (DE-588)4034949-4 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Lebesgue-Integral Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021962335&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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