Measure and integration: a concise introduction to real analysis
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 237 S. graph. Darst. |
| ISBN: | 9780470259542 |
Staff View
MARC
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| 245 | 1 | 0 | |a Measure and integration |b a concise introduction to real analysis |c Leonard F. Richardson |
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2009 | |
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Record in the Search Index
| _version_ | 1804140692063125504 |
|---|---|
| adam_text | CONTENTS
Preface
xi
Acknowledgments
xiii
Introduction
xv
1
History of the Subject
1
1.1
History of the Idea
1
1.2
Deficiencies of the Riemann Integral
3
1.3
Motivation for the Lebesgue Integral
6
2
Fields,
Borei
Fields, and Measures
11
2.1
Fields, Monotone Classes, and
Borei
Fields
11
2.2
Additive Measures
18
2.3
Carathéodory
Outer Measure
20
2.4
E.
Hopfs
Extension Theorem
24
2.4.1
Fields,
σ
-Fields,
and Measures Inherited by a Subset
29
3
Lebesgue Measure
31
vii
VIU
CONTENTS
3.1
The Finite Interval [-N, N)
31
3.2
Measurable Sets,
Borei
Sets, and the Real Line
34
3.2.1
Lebesgue Measure on
R
36
3.3
Measure Spaces and Completions
38
3.3.1
Minimal Completion of a Measure Space
41
3.3.2
A Nonmeasurable Set
41
3.4
Semimetric Space of Measurable Sets
43
3.5
Lebesgue Measure in Rn
50
3.6
Jordan Measure in R
52
Measurable Functions
55
4.1
Measurable Functions
55
4.1.1
Baire Functions of Measurable Functions
56
4.2
Limits of Measurable Functions
58
4.3
Simple Functions and Egoroff s Theorem
61
4.3.1
Double Sequences
63
4.3.2
Convergence in Measure
65
4.4
Lusin s Theorem
66
The Integral
69
5.1
Special Simple Functions
69
5.2
Extending the Domain of the Integral
72
5.2.1
The Class C+ of
Nonnegative
Measurable Functions
74
5.2.2
The Class
С
of Lebesgue
Integrable
Functions
78
5.2.3
Convex Functions and Jensen s Inequality
81
5.3
Lebesgue Dominated Convergence Theorem
83
5.4
Monotone Convergence and Fatou s Theorem
89
5.5
Completeness of LX(X,
21,
μ)
and the Pointwise Convergence
Lemma
92
5.6
Complex-Valued Functions
100
Product Measures and Fubini s Theorem
103
6.1
Product Measures
103
6.2
Fubini s Theorem
108
6.3
Comparison of Lebesgue and Riemann Integrals
117
Functions of a Real Variable
123
CONTENTS
ІХ
7.1
Functions of Bounded Variation
123
7.2
A Fundamental Theorem for the Lebesgue Integral
128
7.3
Lebesgue s Theorem and Vitali s Covering Theorem
131
7.4
Absolutely Continuous and Singular Functions
139
8
General Countably Additive Set Functions
151
8.1 Hahn
Decomposition Theorem
152
8.2
Radon-Nikodym Theorem
156
8.3
Lebesgue Decomposition Theorem
161
9
Examples of Dual Spaces from Measure Theory
165
9.1
TheBanachSpaceLp(X,2l,/i)
165
9.2
The Dual of a Banach Space
170
9.3
The Dual Space of LP(X, SI,
μ)
174
9.4
Hubert Space, Its Dual, and L2{X,
Я,
μ)
178
9.5 Riesz-Markov-Saks-Kakutani
Theorem
185
10
Translation
Invariance
in Real Analysis
195
10.1
An
Orthonormal
Basis for L2(T)
196
10.2
Closed, Invariant Subspaces of
L
2 (T)
203
10.2.1
Integration of Hilbert Space Valued Functions
204
10.2.2
Spectrum of a Subset of
L
2(T)
206
10.3
Schwartz Functions: Fourier Transform and Inversion
208
10.4
Closed, Invariant Subspaces of
L2(R)
213
10.4.1
The Fourier Transform in
2,
2(R)
213
10.4.2
Translation-Invariant Subspaces of
L
2 (R)
216
10.4.3
The Fourier Transform and Direct Integrals
218
10.5
Irreducibility of £2(R) Under Translations and Rotations
219
10.5.1
Position and Momentum Operators
221
10.5.2
The
Heisenberg
Group
222
Appendix: The Banach-Tarski Theorem
225
A.I The Limits to Countable Additivity
225
References
229
Index
231
|
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| id | DE-604.BV035765501 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:04:01Z |
| institution | BVB |
| isbn | 9780470259542 |
| language | English |
| lccn | 2009009714 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018625303 |
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| physical | XVI, 237 S. graph. Darst. |
| publishDate | 2009 |
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| publisher | Wiley |
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| spelling | Richardson, Leonard F. 1944- Verfasser (DE-588)136033164 aut Measure and integration a concise introduction to real analysis Leonard F. Richardson Hoboken, NJ Wiley 2009 XVI, 237 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lebesgue integral Measure theory Mathematical analysis 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 Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018625303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Richardson, Leonard F. 1944- Measure and integration a concise introduction to real analysis Lebesgue integral Measure theory Mathematical analysis 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 | Measure and integration a concise introduction to real analysis |
| title_auth | Measure and integration a concise introduction to real analysis |
| title_exact_search | Measure and integration a concise introduction to real analysis |
| title_full | Measure and integration a concise introduction to real analysis Leonard F. Richardson |
| title_fullStr | Measure and integration a concise introduction to real analysis Leonard F. Richardson |
| title_full_unstemmed | Measure and integration a concise introduction to real analysis Leonard F. Richardson |
| title_short | Measure and integration |
| title_sort | measure and integration a concise introduction to real analysis |
| title_sub | a concise introduction to real analysis |
| topic | Lebesgue integral Measure theory Mathematical analysis Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Lebesgue integral Measure theory Mathematical analysis Integrationstheorie Maßtheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018625303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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