Measure, integration and function spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
1994
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 277 S. |
| ISBN: | 9810216106 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | TABLE OF CONTENTS
Preface vii
1. Introduction 1
1.1 Preliminaries 1
1.2 Extended Real Numbers (If) and R 4
1.3 Lebesgue s Definition of the Integral 9
2. Measure Theory 15
2.1 Semi rings and Algebras of Sets 15
2.2 Additive Set Functions 20
2.2.1 Jordan Decomposition 28
2.2.2 Hahn Decomposition 33
2.2.3 Drewnowski s Lemma 35
2.3 Outer Measures 36
2.3.1 Metric Outer Measures 39
2.4 Extensions of Premeasures 41
2.4.1 Hewitt Yosida Decomposition 49
2.5 Lebesgue Measure 50
2.6 Lebesgue Stieltjes Measure 57
2.6.1 Hewitt Yosida Decomposition for 60
Lebesgue Stieltjes Measures
2.7 Regular Measures 62
2.8 The Nikodym Convergence and Boundedness Theorems 66
3. Integration 71
3.1 Measurable Functions 71
3.1.1 Approximation of Measurable Functions 78
3.2 The Lebesgue Integral 81
3.3 The Riemann and Lebesgue Integrals 96
3.4 Integrals Depending on a Parameter 100
3.5 Convergence in Mean 103
3.6 Convergence in Measure 107
3.7 Comparison of Modes of Convergence 112
3.8 Mikusinski s Characterization of the Lebesgue Measure 115
3.9 Product Measures and Fubini s Theorem 118
3.10 A Geometric Interpretation of the Integral 126
3.11 Convolution Product 127
3.12 The Radon Nikodym Theorem 132
3.12.1 The Radon Nikodym Theorem for Finitely 137
Additive Set Functions
3.13 Lebesgue Decomposition 141
3.14 The Vitali Hahn Saks Theorem 144
ix
x CONTENTS
4. Differentiation and Integration 147
4.1 Differentiating Indefinite Integrals 147
4.2 Differentiation of Monotone Functions 152
4.3 Integrating Derivatives 158
4.4 Absolutely Continuous Functions 160
5. Introduction to Functional Analysis 165
5.1 Normed Linear Spaces (NLS) 165
5.2 Linear Mappings between Normed Linear Spaces 174
5.3 The Uniform Boundedness Principle 178
5.4 Quotient Spaces 181
5.5 The Closed Graph/Open Mapping Theorems 183
5.6 The Hahn Banach Theorem 186
5.6.1 Applications of the Hahn Banach Theorem in NLS 189
5.6.2 Extension of Bounded, Finitely Additive Set Functions 193
5.6.3 A Translation Invariant, Finitely Additive Set Function 194
5.7 Ordered Linear Spaces 197
6. Function Spaces 207
6.1 L spaces, 1 p oo 207
6.2 The Space L°°{n) 218
6.3 The Space of Finitely Additive Set Functions 223
6.4 The Space of Countably Additive Set Functions 226
6.5 The Space of Continuous Functions 228
6.6 Hilbert Space 235
6.6.1 Fourier Transform 249
Appendix 255
Al. Functions of Bounded Variation 255
A2. The Baire Category Theorem 258
A3. The Arzela Ascoli Theorem 260
A4. The Stone Weierstrass Theorem 262
References 267
Index 272
|
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| indexdate | 2024-07-09T17:53:19Z |
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| isbn | 9810216106 |
| language | English |
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| publishDate | 1994 |
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| publisher | World Scientific |
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| spelling | Swartz, Charles 1938- Verfasser (DE-588)131653601 aut Measure, integration and function spaces Charles Swartz Singapore [u.a.] World Scientific 1994 X, 277 S. txt rdacontent n rdamedia nc rdacarrier Espaces fonctionnels ram Intégrales généralisées ram Mesure, Théorie de la ram Function spaces Integrals, Generalized Measure theory Integration Mathematik (DE-588)4072852-3 gnd rswk-swf Funktionenraum (DE-588)4134834-5 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Integration Mathematik (DE-588)4072852-3 s Funktionenraum (DE-588)4134834-5 s DE-604 Maßtheorie (DE-588)4074626-4 s Integrationstheorie (DE-588)4138369-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006987854&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Swartz, Charles 1938- Measure, integration and function spaces Espaces fonctionnels ram Intégrales généralisées ram Mesure, Théorie de la ram Function spaces Integrals, Generalized Measure theory Integration Mathematik (DE-588)4072852-3 gnd Funktionenraum (DE-588)4134834-5 gnd Maßtheorie (DE-588)4074626-4 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| subject_GND | (DE-588)4072852-3 (DE-588)4134834-5 (DE-588)4074626-4 (DE-588)4138369-2 |
| title | Measure, integration and function spaces |
| title_auth | Measure, integration and function spaces |
| title_exact_search | Measure, integration and function spaces |
| title_full | Measure, integration and function spaces Charles Swartz |
| title_fullStr | Measure, integration and function spaces Charles Swartz |
| title_full_unstemmed | Measure, integration and function spaces Charles Swartz |
| title_short | Measure, integration and function spaces |
| title_sort | measure integration and function spaces |
| topic | Espaces fonctionnels ram Intégrales généralisées ram Mesure, Théorie de la ram Function spaces Integrals, Generalized Measure theory Integration Mathematik (DE-588)4072852-3 gnd Funktionenraum (DE-588)4134834-5 gnd Maßtheorie (DE-588)4074626-4 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| topic_facet | Espaces fonctionnels Intégrales généralisées Mesure, Théorie de la Function spaces Integrals, Generalized Measure theory Integration Mathematik Funktionenraum Maßtheorie Integrationstheorie |
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