Introduction to the theory of integration:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Acad. Press
1971
|
| Edition: | 2. print. |
| Series: | Pure and applied mathematics
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 385 S. |
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MARC
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| 100 | 1 | |a Hildebrandt, Theophil H. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to the theory of integration |c T. H. Hildebrandt |
| 250 | |a 2. print. | ||
| 264 | 1 | |a New York [u.a.] |b Acad. Press |c 1971 | |
| 300 | |a IX, 385 S. | ||
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| 490 | 1 | |a Pure and applied mathematics |v 13 | |
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Record in the Search Index
| _version_ | 1804128570587480064 |
|---|---|
| adam_text | CONTENTS
Preface v
Chapter I
A General Theory of Limits
1. Directed Sets. The Moore Smith General Limit 1
2. Properties of Limits 5
3. Values Approached 7
4. Extreme Limits 9
5. Directed Sets of Sequential Character 11
6. General Sums 12
7. Double and Iterated Limits 13
8. Filters 17
9. Linear Spaces 20
Chapter II
Riemannian Type of Integration
/. Functions of Intervals 25
2. Integrals of Functions of Intervals 27
3. Existence Theorems 28
4. The Integral of an Interval Function as a Function of Intervals .... 33
5. Integral as a Function of the Interval Function 35
6. Functions of Bounded Variation 37
7. Continuity Properties of Functions of Bounded Variation 40
8. The Space of Functions of Bounded Variation 41
9. Riemann Stieltjes Integrals 47
10. Existence Theorems for Riemann Stieltjes Integrals 49
11. Properties of Riemann Stieltjes Integrals 52
12. Classes of Functions Determined by Stieltjes Integrals 56
13. Riemann Stieltjes Integrals with Respect to Functions of Bounded Var¬
iation. Existence Theorems 58
14. Properties of J* fdg with g of Bounded Variation on (a, b) 67
15. Convergence Theorems 69
16. The Integral as Function of the Upper Limit 76
17. The Integral as a Function of a Parameter 78
18. Linear Continuous Functionals on the Space of Continuous Functions . . 81
19. Additional Definitions of Riemann Integrals of Stieltjes Type 86
20. Stieltjes Integrals on Infinite Intervals 97
21. A Linear Form on the Infinite Interval 98
Chapter III
Integrals of Riemann Type of Functions of Intervals in Two
or Higher Dimension
/. Interval Functions 101
2. Subdivisions 102
3. Integrals 103
4. Functions of Bounded Variation in Two Dimensions 106
5. Continuity Properties 109
vii
yiii CONTENTS
6. The Space of Functions of Bounded Variation in Two Variables: BV2. . . 114
7. Functions of Bounded Variation According to Frechet 116
8. Riemann Stieltjes Integrals in Two Variables 123
9. Stieltjes Integrals in Two Dimensions with Respect to Functions of Bounded
Variation 128
10. Relation between Double and Iterated Integrals 132
11. Double Integrals with Respect to Functions of Frechet Bounded Variation 136
Chapter IV
Sets
1. Fundamental Operations 141
2. Characteristic Functions 146
3. Properties of Classes of Sets 146
4. Extensions of Classes of Sets Relative to Properties 148
Chapter V
Content and Measure
/. Content of a Linear Bounded Set 155
2. Properties of the Content Functions 157
3. Borel Measurability 162
4. General Lebesgue Measure of Linear Sets 163
5. Lower Measure 166
6. Measurability 168
7. Examples of Measurable Sets 171
8. Sets of Zero Measure 172
9. Additional Properties of Upper and Lower Measure 173
10. Additional Conditions for Measurability 173
//. The Function a(x) Is Unbounded 175
12. Lebesgue Measure 179
13. Relation between a Measure and Lebesgue Measure 182
14. Relations between Classes of a Measurable Sets 183
15. Measurability of Sets in Euclidean Space of Higher Dimension 188
16. Some Abstract Measure Theory 191
17. Upper Semiadditive Functions 196
Chapter VI
Measurable Functions
/. Semicontinuous Functions 199
2. Measurable Functions 203
3. Properties of Measurable Functions 205
4. Examples of Measurable Functions 207
5. Properties of Measurable Functions Depending on the Measure Functions 209
6. Approximations to Measurable Functions. Lusin s Theorem 215
Chapter VII
Lebesgue Stieltjes Integration
/. The Lebesgue Postulates on Integration 219
2. The Lebesgue Method of Defining an Integral 224
3. A Riemann Young Type of Integral Definition. The K Integral 231
CONTENTS ix
4. Properties of the Integrals 239
5. The Integral as a Function of Measurable Sets 239
6. L S Integrals as Functions of Intervals 241
7. Absolute Continuity 242
8. Properties of the Integral /(/, a, E) = Efda as a Function of/ .... 244
9. Directed Limits and L S Integration 255
10. Properties of the Integral /(/, a, E) = Efda as a Function of a . . . 258
11. Integration with Respect to Functions of Bounded Variation 261
12. Integration with Respect to Unbounded Measure Functions 274
Chapter VIII
Classes of Measurable and Integrable Functions
/. The Class of cc Measurable Functions 285
2. The Class of Almost Bounded a Measurable Functions 288
3. The Space Lx 290
4. The Space W with 0 p oo 291
5. Separability 296
6. The Space L2. Orthogonal Functions. Riesz Fischer Theorem 299
Chapter IX
Other Methods of Defining the Class of Lebesgue Integrable
Functions. Abstract Integrals
/. The Space L1 as the Completion of a Metric Space by Cauchy Sequences 310
2. Construction of the Space L1 by the Use of Osgood s Theorem 312
3. L S Integration Based on Monotonic Sequences of Semicontinuous Functions 317
4. Lebesgue Stieltjes Integrals on an Abstract Set 320
Chapter X
Product Measures. Iterated Integrals. Fubini Theorem
/. Product Measures 327
2. The Lebesgue Stieltjes Integrals as the Measure of a Plane Set 333
3. Fubini Theorem on Double and Iterated Integrals 335
Chapter XI
Derivatives and Integrals
1. Riemann Integrals and Derivatives 343
2. On Derivatives 345
3. The Vitali Theorem 354
4. Derivatives of Monotonic Functions and of Functions of Bounded Variation 357
5. Derivatives of Indefinite Lebesgue Integrals 361
6. Lebesgue Integrals of Derivatives 365
7. Lebesgue Decomposition of Functions of Bounded Variation 369
8. The Radon Nikodym Theorem 375
Some Reference Books on Integration 379
Subject Index 381
|
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| author_sort | Hildebrandt, Theophil H. |
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| bvnumber | BV013750717 |
| classification_rvk | SK 430 |
| ctrlnum | (OCoLC)256302791 (DE-599)BVBBV013750717 |
| discipline | Mathematik |
| edition | 2. print. |
| format | Book |
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| indexdate | 2024-07-09T18:51:21Z |
| institution | BVB |
| language | English |
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| physical | IX, 385 S. |
| publishDate | 1971 |
| publishDateSearch | 1971 |
| publishDateSort | 1971 |
| publisher | Acad. Press |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Hildebrandt, Theophil H. Verfasser aut Introduction to the theory of integration T. H. Hildebrandt 2. print. New York [u.a.] Acad. Press 1971 IX, 385 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 13 Integration Mathematik (DE-588)4072852-3 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 s DE-604 Integration Mathematik (DE-588)4072852-3 s Pure and applied mathematics 13 (DE-604)BV010177228 13 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399583&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hildebrandt, Theophil H. Introduction to the theory of integration Pure and applied mathematics Integration Mathematik (DE-588)4072852-3 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| subject_GND | (DE-588)4072852-3 (DE-588)4138369-2 |
| title | Introduction to the theory of integration |
| title_auth | Introduction to the theory of integration |
| title_exact_search | Introduction to the theory of integration |
| title_full | Introduction to the theory of integration T. H. Hildebrandt |
| title_fullStr | Introduction to the theory of integration T. H. Hildebrandt |
| title_full_unstemmed | Introduction to the theory of integration T. H. Hildebrandt |
| title_short | Introduction to the theory of integration |
| title_sort | introduction to the theory of integration |
| topic | Integration Mathematik (DE-588)4072852-3 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| topic_facet | Integration Mathematik Integrationstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009399583&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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