A radical approach to Lebesgue's theory of integration:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2008
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | MAA textbooks
|
| Schlagworte: | |
| Online-Zugang: | Table of contents only Publisher description Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 317-322) and index |
| Beschreibung: | xiv, 329 p. ill. 26 cm |
| ISBN: | 9780521884747 0521884748 9780521711838 0521711835 |
Internformat
MARC
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| 245 | 1 | 0 | |a A radical approach to Lebesgue's theory of integration |c David M. Bressoud |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2008 | |
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Datensatz im Suchindex
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| adam_text | Contents
ice page xi
Introduction 1
1.1 The Five Big Questions 2
1.2 Presumptions 15
The Riemann Integral 23
2.1 Existence 24
2.2 Nondifferentiable Integrals 33
2.3 The Class of 1870 40
Explorations of R 51
3.1 Geometry of K 51
3.2 Accommodating Algebra 59
3.3 Set Theory 71
Nowhere Dense Sets and the Problem with the Fundamental
Theorem of Calculus 81
4.1 The Smith-Volterra-Cantor Sets 82
4.2 Volterra s Function 89
4.3 Term-by-Term Integration 98
4.4 The Baire Category Theorem 109
The Development of Measure Theory 120
5.1 Peano, Jordan, and Borel 122
5.2 Lebesgue Measure 131
5.3 Carathéodory s Condition 140
5.4 Nonmeasurable Sets 150
The Lebesgue Integral 159
6.1 Measurable Functions 159
6.2 Integration 169
6.3 Lebesgue s Dominated Convergence Theorem 183
6.4 Egorov s Theorem 191
IX
x Contents
7 The Fundamental Theorem of Calculus 203
7.1 The Dini Derivatives 204
7.2 Monotonicity Implies Differentiability Almost Everywhere 212
7.3 Absolute Continuity 223
7.4 Lebesgue s FTC 231
8 Fourier Series 241
8.1 Pointwise Convergence 242
8.2 Metric Spaces 251
8.3 Banach Spaces 263
8.4 Hilbert Spaces 271
9 Epilogue 282
Appendixes 287
A Other Directions 287
A. 1 The Cardinality of the Collection of Borel Sets 287
A.2 The Generalized Riemann Integral 291
B Hints to Selected Exercises 299
Bibliography 317
Index 323
|
| adam_txt |
Contents
ice page xi
Introduction 1
1.1 The Five Big Questions 2
1.2 Presumptions 15
The Riemann Integral 23
2.1 Existence 24
2.2 Nondifferentiable Integrals 33
2.3 The Class of 1870 40
Explorations of R 51
3.1 Geometry of K 51
3.2 Accommodating Algebra 59
3.3 Set Theory 71
Nowhere Dense Sets and the Problem with the Fundamental
Theorem of Calculus 81
4.1 The Smith-Volterra-Cantor Sets 82
4.2 Volterra's Function 89
4.3 Term-by-Term Integration 98
4.4 The Baire Category Theorem 109
The Development of Measure Theory 120
5.1 Peano, Jordan, and Borel 122
5.2 Lebesgue Measure 131
5.3 Carathéodory's Condition 140
5.4 Nonmeasurable Sets 150
The Lebesgue Integral 159
6.1 Measurable Functions 159
6.2 Integration 169
6.3 Lebesgue's Dominated Convergence Theorem 183
6.4 Egorov's Theorem 191
IX
x Contents
7 The Fundamental Theorem of Calculus 203
7.1 The Dini Derivatives 204
7.2 Monotonicity Implies Differentiability Almost Everywhere 212
7.3 Absolute Continuity 223
7.4 Lebesgue's FTC 231
8 Fourier Series 241
8.1 Pointwise Convergence 242
8.2 Metric Spaces 251
8.3 Banach Spaces 263
8.4 Hilbert Spaces 271
9 Epilogue 282
Appendixes 287
A Other Directions 287
A. 1 The Cardinality of the Collection of Borel Sets 287
A.2 The Generalized Riemann Integral 291
B Hints to Selected Exercises 299
Bibliography 317
Index 323 |
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| isbn | 9780521884747 0521884748 9780521711838 0521711835 |
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| spelling | Bressoud, David M. 1950- Verfasser (DE-588)136747221 aut A radical approach to Lebesgue's theory of integration David M. Bressoud 1. publ. Cambridge Cambridge University Press 2008 xiv, 329 p. ill. 26 cm txt rdacontent n rdamedia nc rdacarrier MAA textbooks Includes bibliographical references (p. 317-322) and index Integrals, Generalized Lebesgue-Integral (DE-588)4034949-4 gnd rswk-swf Lebesgue-Integral (DE-588)4034949-4 s DE-604 http://www.loc.gov/catdir/enhancements/fy0803/2007035326-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0803/2007035326-d.html Publisher description HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016532837&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bressoud, David M. 1950- A radical approach to Lebesgue's theory of integration Integrals, Generalized Lebesgue-Integral (DE-588)4034949-4 gnd |
| subject_GND | (DE-588)4034949-4 |
| title | A radical approach to Lebesgue's theory of integration |
| title_auth | A radical approach to Lebesgue's theory of integration |
| title_exact_search | A radical approach to Lebesgue's theory of integration |
| title_exact_search_txtP | A radical approach to Lebesgue's theory of integration |
| title_full | A radical approach to Lebesgue's theory of integration David M. Bressoud |
| title_fullStr | A radical approach to Lebesgue's theory of integration David M. Bressoud |
| title_full_unstemmed | A radical approach to Lebesgue's theory of integration David M. Bressoud |
| title_short | A radical approach to Lebesgue's theory of integration |
| title_sort | a radical approach to lebesgue s theory of integration |
| topic | Integrals, Generalized Lebesgue-Integral (DE-588)4034949-4 gnd |
| topic_facet | Integrals, Generalized Lebesgue-Integral |
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