The generalized Riemann integral:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Washington, DC]
Math. Assoc. of America
1980
|
| Ausgabe: | 1. print. |
| Schriftenreihe: | The Carus mathematical monographs
20 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 275 S. |
| ISBN: | 0883850214 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804136070460211200 |
|---|---|
| adam_text | CONTENTS
PAGE
Preface vii
List of Symbols ix
Introduction 1
Chapter 1—Definition of the Generalized
Riemann Integral 5
1.1 Selecting Riemann sums 7
1.2 Definition of the generalized Riemann in¬
tegral 17
1.3 Integration over unbounded intervals 21
1.4 The fundamental theorem of calculus 25
1.5 The status of improper integrals 28
1.6 Multiple integrals 31
1.7 Sum of a series viewed as an integral 36
51.8 The limit based on gauges 38
51.9 Proof of the fundamental theorem 40
1.10 Exercises 45
Chapter 2—Basic Properties of the Integral 47
2.1 The integral as a function of the integrand 48
2.2 The Cauchy criterion 50
2.3 Integrability on subintervals 51
2.4 The additivity of integrals 52
2.5 Finite additivity of functions of intervals 55
2.6 Continuity of integrals. Existence of
primitives 58
2.7 Change of variables in integrals on inter¬
vals in R 59
S2.8 Limits of integrals over expanding
intervals 65
2.9 Exercises 68
xi
Xii CONTENTS
Chapter 3—Absolute Integrability and Con¬
vergence Theorems 71
3.1 Henstock s lemma 74
3.2 Integrability of the absolute value of an
integrable function 77
3.3 Lattice operations on integrable functions 81
3.4 Uniformly convergent sequences of func¬
tions 83
3.5 The monotone convergence theorem 86
3.6 The dominated convergence theorem 88
53.7 Proof of Henstock s lemma 90
53.8 Proof of the criterion for integrability
of |/| 91
53.9 Iterated limits 93
53.10 Proof of the monotone and dominated
convergence theorems 96
3.11 Exercises 101
Chapter 4—Integration on Subsets of
Intervals 103
4.1 Null functions and null sets 104
4.2 Convergence almost everywhere 110
4.3 Integration over sets which are not inter¬
vals 113
4.4 Integration of continuous functions on
closed, bounded sets 116
4.5 Integrals on sequences of sets 118
4.6 Length, area, volume, and measure 122
4.7 Exercises 128
Chapter 5—Measurable Functions 131
S.I Measurable functions 133
52 Measurability and absolute integrability 135
5.3 Operations on measurable functions 139
5.4 Integrability of products 141
S5.5 Approximation by step functions 143
5.6 Exercises 147
Chapter 6—Multiple and Iterated Integrals 149
6.1 Fubini s theorem 150
62 Determining integrability from iterated
integrals 154
CONTENTS Xiii
56.3 Compound divisions. Compatibility
theorem 164
56.4 Proof of Fubini s theorem 168
56.5 Double series 171
6.6 Exercises 173
Chapter 7—Integrals of Stieltjes Type 177
7.1 Three versions of the Riemann Stieltjes
integral 180
7.2 Basic properties of Riemann Stieltjes
integrals 183
13 Limits, continuity, and differentiability of
integrals 187
7.4 Values of certain integrals 189
7.5 Existence theorems for Riemann Stieltjes
integrals 192
7.6 Integration by parts 195
7.7 Integration of absolute values. Lattice
operations 200
7.8 Monotone and dominated convergence 204
7.9 Change of variables 205
7.10 Mean value theorems for integrals 209
57.11 Sequences of integrators ° 211
57.12 Line integrals 214
57.13 Functions of bounded variation and reg¬
ulated functions 220
57.14 Proof of the absolute integrability
theorem 225
7.15 Exercises 227
Chapter 8—Comparison of Integrals 231
58.1 Characterization of measurable sets 232
58.2 Lebesgue measure and integral 234
S83 Characterization of absolute integrability
using Riemann sums 236
8.4 Suggestions for further study 244
References 245
Appendix Solutions of In text Exercises 247
Index 269
|
| adam_txt |
CONTENTS
PAGE
Preface vii
List of Symbols ix
Introduction 1
Chapter 1—Definition of the Generalized
Riemann Integral 5
1.1 Selecting Riemann sums 7
1.2 Definition of the generalized Riemann in¬
tegral 17
1.3 Integration over unbounded intervals 21
1.4 The fundamental theorem of calculus 25
1.5 The status of improper integrals 28
1.6 Multiple integrals 31
1.7 Sum of a series viewed as an integral 36
51.8 The limit based on gauges 38
51.9 Proof of the fundamental theorem 40
1.10 Exercises 45
Chapter 2—Basic Properties of the Integral 47
2.1 The integral as a function of the integrand 48
2.2 The Cauchy criterion 50
2.3 Integrability on subintervals 51
2.4 The additivity of integrals 52
2.5 Finite additivity of functions of intervals 55
2.6 Continuity of integrals. Existence of
primitives 58
2.7 Change of variables in integrals on inter¬
vals in R 59
S2.8 Limits of integrals over expanding
intervals 65
2.9 Exercises 68
xi
Xii CONTENTS
Chapter 3—Absolute Integrability and Con¬
vergence Theorems 71
3.1 Henstock's lemma 74
3.2 Integrability of the absolute value of an
integrable function 77
3.3 Lattice operations on integrable functions 81
3.4 Uniformly convergent sequences of func¬
tions 83
3.5 The monotone convergence theorem 86
3.6 The dominated convergence theorem 88
53.7 Proof of Henstock's lemma 90
53.8 Proof of the criterion for integrability
of |/| 91
53.9 Iterated limits 93
53.10 Proof of the monotone and dominated
convergence theorems 96
3.11 Exercises 101
Chapter 4—Integration on Subsets of
Intervals 103
4.1 Null functions and null sets 104
4.2 Convergence almost everywhere 110
4.3 Integration over sets which are not inter¬
vals 113
4.4 Integration of continuous functions on
closed, bounded sets 116
4.5 Integrals on sequences of sets 118
4.6 Length, area, volume, and measure 122
4.7 Exercises 128
Chapter 5—Measurable Functions 131
S.I Measurable functions 133
52 Measurability and absolute integrability 135
5.3 Operations on measurable functions 139
5.4 Integrability of products 141
S5.5 Approximation by step functions 143
5.6 Exercises 147
Chapter 6—Multiple and Iterated Integrals 149
6.1 Fubini's theorem 150
62 Determining integrability from iterated
integrals 154
CONTENTS Xiii
56.3 Compound divisions. Compatibility
theorem 164
56.4 Proof of Fubini's theorem 168
56.5 Double series 171
6.6 Exercises 173
Chapter 7—Integrals of Stieltjes Type 177
7.1 Three versions of the Riemann Stieltjes
integral 180
7.2 Basic properties of Riemann Stieltjes
integrals 183
13 Limits, continuity, and differentiability of
integrals 187
7.4 Values of certain integrals 189
7.5 Existence theorems for Riemann Stieltjes
integrals 192
7.6 Integration by parts 195
7.7 Integration of absolute values. Lattice
operations 200
7.8 Monotone and dominated convergence 204
7.9 Change of variables 205
7.10 Mean value theorems for integrals 209
57.11 Sequences of integrators ° 211
57.12 Line integrals 214
57.13 Functions of bounded variation and reg¬
ulated functions 220
57.14 Proof of the absolute integrability
theorem 225
7.15 Exercises 227
Chapter 8—Comparison of Integrals 231
58.1 Characterization of measurable sets 232
58.2 Lebesgue measure and integral 234
S83 Characterization of absolute integrability
using Riemann sums 236
8.4 Suggestions for further study 244
References 245
Appendix Solutions of In text Exercises 247
Index 269 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:14:56Z |
| indexdate | 2024-07-09T20:50:33Z |
| institution | BVB |
| isbn | 0883850214 |
| language | English |
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| physical | XIII, 275 S. |
| publishDate | 1980 |
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| series | The Carus mathematical monographs |
| series2 | The Carus mathematical monographs |
| spelling | MacLeod, Robert M. Verfasser aut The generalized Riemann integral by Robert M. McLeod 1. print. [Washington, DC] Math. Assoc. of America 1980 XIII, 275 S. txt rdacontent n rdamedia nc rdacarrier The Carus mathematical monographs 20 Riemann, Intégrale de Riemann-integralen gtt Teoria Da Medida larpcal Riemann integral Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Integralrechnung (DE-588)4027232-1 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Integralrechnung (DE-588)4027232-1 s DE-604 Riemannsche Fläche (DE-588)4049991-1 s Maßtheorie (DE-588)4074626-4 s 1\p DE-604 Integrationstheorie (DE-588)4138369-2 s 2\p DE-604 The Carus mathematical monographs 20 (DE-604)BV001887236 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310254&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | MacLeod, Robert M. The generalized Riemann integral The Carus mathematical monographs Riemann, Intégrale de Riemann-integralen gtt Teoria Da Medida larpcal Riemann integral Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd Integralrechnung (DE-588)4027232-1 gnd Riemannsche Fläche (DE-588)4049991-1 gnd |
| subject_GND | (DE-588)4138369-2 (DE-588)4074626-4 (DE-588)4027232-1 (DE-588)4049991-1 |
| title | The generalized Riemann integral |
| title_auth | The generalized Riemann integral |
| title_exact_search | The generalized Riemann integral |
| title_exact_search_txtP | The generalized Riemann integral |
| title_full | The generalized Riemann integral by Robert M. McLeod |
| title_fullStr | The generalized Riemann integral by Robert M. McLeod |
| title_full_unstemmed | The generalized Riemann integral by Robert M. McLeod |
| title_short | The generalized Riemann integral |
| title_sort | the generalized riemann integral |
| topic | Riemann, Intégrale de Riemann-integralen gtt Teoria Da Medida larpcal Riemann integral Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd Integralrechnung (DE-588)4027232-1 gnd Riemannsche Fläche (DE-588)4049991-1 gnd |
| topic_facet | Riemann, Intégrale de Riemann-integralen Teoria Da Medida Riemann integral Integrationstheorie Maßtheorie Integralrechnung Riemannsche Fläche |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310254&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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