Measures, integrals and martingales:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ., reprint. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 381 S. graph. Darst. |
| ISBN: | 9780521850155 9780521615259 0521615259 0521850150 |
Staff View
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| 100 | 1 | |a Schilling, René L. |d 1969- |e Verfasser |0 (DE-588)122050045 |4 aut | |
| 245 | 1 | 0 | |a Measures, integrals and martingales |c René L. Schilling |
| 250 | |a 1. publ., reprint. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
| 300 | |a X, 381 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804137787487682560 |
|---|---|
| adam_text | Contents
Prelude
Dependence chart
1
Prologue
Problems
2
The pleasures of counting
Problems
3
σ
-algebras
Problems
4
Measures
Problems
5
Uniqueness of measures
Problems
6
Existence of measures
Problems
7
Measurable mappings
Problems
8
Measurable functions
Problems
page
viii
xi
1
4
5
13
15
20
22
28
31
35
37
46
49
54
57
65
9
Integration of positive functions
67
Problems
73
10
Integrals of measurable functions and null sets
76
Null sets and the a-e.
80
Problems
84
vi
Contents
11
Convergence
theorems and their applications
88
Parameter-dependent integrals
91
Riemann vs. Lebesgue integration
92
Examples
98
Problems
100
12
The function spaces £P
, 1
^p
«ζ οο
105
Problems
116
13
Product measures and Fubini s theorem
120
More on measurable functions
127
Distribution functions
128
Minkowski s inequality for integrals
130
Problems
130
14
Integrals with respect to image measures
134
Convolutions
137
Problems
140
15
Integrals of images and Jacobi s transformation rule
142
Jacobi s transformation formula
147
Spherical coordinates and the volume of the unit ball
152
Continuous functions are dense in ^{k )
156
Regular measures
158
Problems
159
16
Uniform integrability and Vitali s convergence theorem
163
Different forms of uniform integrability
168
Problems
173
17
Martingales
176
Problems
188
18
Martingale convergence theorems
190
Problems
200
19
The
Radon-Nikodým
theorem and other applications
of martingales
202
The
Radon-Nikodým
theorem
202
Martingale inequalities
211
The Hardy-Littlewood maximal theorem
213
Lebesgue s differentiation theorem
218
The
Calderón-Zygmund
lemma
221
Problems
222
Contents
vii
20 Inner
product
spaces
226
Problems
232
21
Hilbert space
£ 234
Problems
246
22
Conditional expectations in L2
248
On the structure of subspaces of L2
253
Problems
257
23
Conditional expectations in LP
258
Classical conditional expectations
263
Separability criteria for the spaces LP{X,
Л,
μ)
269
Problems
274
24
Orthonormal
systems and their convergence behaviour
276
Orthogonal polynomials
276
The trigonometric system and Fourier series
283
The
Haar
system
289
The
Haar
wavelet
295
The Rademacher functions
299
Well-behaved
orthonormal
systems
302
Problems
312
Appendix
A: lim inf
and
lim sup
313
Appendix B: Some facts from point-set topology
318
Topological spaces
319
Metric spaces
322
Normed spaces
325
Appendix C: The volume of a parallelepiped
328
Appendix D: Non-measurable sets
330
Appendix
E: A
summary of the Riemann integral
337
The (proper) Riemann integral
337
The fundamental theorem of integral calculus
346
Integrals and limits
351
Improper Riemann integrals
353
Further reading
360
References
364
Notation index
367
Name and subject index
37
1
|
| adam_txt |
Contents
Prelude
Dependence chart
1
Prologue
Problems
2
The pleasures of counting
Problems
3
σ
-algebras
Problems
4
Measures
Problems
5
Uniqueness of measures
Problems
6
Existence of measures
Problems
7
Measurable mappings
Problems
8
Measurable functions
Problems
page
viii
xi
1
4
5
13
15
20
22
28
31
35
37
46
49
54
57
65
9
Integration of positive functions
67
Problems
73
10
Integrals of measurable functions and null sets
76
Null sets and the 'a-e.'
80
Problems
84
vi
Contents
11
Convergence
theorems and their applications
88
Parameter-dependent integrals
91
Riemann vs. Lebesgue integration
92
Examples
98
Problems
100
12
The function spaces £P
, 1
^p
«ζ οο
105
Problems
116
13
Product measures and Fubini's theorem
120
More on measurable functions
127
Distribution functions
128
Minkowski's inequality for integrals
130
Problems
130
14
Integrals with respect to image measures
134
Convolutions
137
Problems
140
15
Integrals of images and Jacobi's transformation rule
142
Jacobi's transformation formula
147
Spherical coordinates and the volume of the unit ball
152
Continuous functions are dense in ^{k")
156
Regular measures
158
Problems
159
16
Uniform integrability and Vitali's convergence theorem
163
Different forms of uniform integrability
168
Problems
173
17
Martingales
176
Problems
188
18
Martingale convergence theorems
190
Problems
200
19
The
Radon-Nikodým
theorem and other applications
of martingales
202
The
Radon-Nikodým
theorem
202
Martingale inequalities
211
The Hardy-Littlewood maximal theorem
213
Lebesgue's differentiation theorem
218
The
Calderón-Zygmund
lemma
221
Problems
222
Contents
vii
20 Inner
product
spaces
226
Problems
232
21
Hilbert space
£ 234
Problems
246
22
Conditional expectations in L2
248
On the structure of subspaces of L2
253
Problems
257
23
Conditional expectations in LP
258
Classical conditional expectations
263
Separability criteria for the spaces LP{X,
Л,
μ)
269
Problems
274
24
Orthonormal
systems and their convergence behaviour
276
Orthogonal polynomials
276
The trigonometric system and Fourier series
283
The
Haar
system
289
The
Haar
wavelet
295
The Rademacher functions
299
Well-behaved
orthonormal
systems
302
Problems
312
Appendix
A: lim inf
and
lim sup
313
Appendix B: Some facts from point-set topology
318
Topological spaces
319
Metric spaces
322
Normed spaces
325
Appendix C: The volume of a parallelepiped
328
Appendix D: Non-measurable sets
330
Appendix
E: A
summary of the Riemann integral
337
The (proper) Riemann integral
337
The fundamental theorem of integral calculus
346
Integrals and limits
351
Improper Riemann integrals
353
Further reading
360
References
364
Notation index
367
Name and subject index
37
1 |
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| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
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| spelling | Schilling, René L. 1969- Verfasser (DE-588)122050045 aut Measures, integrals and martingales René L. Schilling 1. publ., reprint. Cambridge [u.a.] Cambridge Univ. Press 2007 X, 381 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Integrals Martingales (Mathematics) Measure theory 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 1\p DE-604 DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016586291&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Schilling, René L. 1969- Measures, integrals and martingales Integrals Martingales (Mathematics) Measure theory 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_exact_search_txtP | 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 | Integrals Martingales (Mathematics) Measure theory Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Integrals Martingales (Mathematics) Measure theory Integrationstheorie Maßtheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016586291&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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