An introduction to integration and measure theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1997
|
| Series: | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts
[16] |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 473 S. |
| ISBN: | 0471595187 |
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MARC
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Record in the Search Index
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| adam_text | Contents
PART ONE LIMIT A TIONS OF THE RIEMANN INTEGRAL 1
Chapter 1 Limits of Integrals and Integrability 3
1.1 General Discussion of the Problem 3
1.2 Examples of Nonintegrability 6
1.3 Examples of Limits of Integrals 7
1.4 Examples of Incompleteness of Norms 8
Exercises 9
Chapter 2 Expectations in Probability Theory 11
2.1 Probabilities 11
2.2 Distributions and Expectations 12
2.3 Two Examples 14
Exercises 15
PART TWO RIEMANN STIELTJES INTEGRALS 17
Chapter 3 Riemann Stieltjes Integrals: Introduction 19
3.1 Partitions 19
3.2 Riemann Stieltjes and Darboux Stieltjes Sums 20
3.3 Riemann Stieltjes Integrals 22
3.4 Some Examples 25
3.5 Properties of Riemann Stieltjes Integrals 27
3.6 Another Example 32
Exercises 34
Chapter 4 Characterization of Riemann Stieltjes Integrability 38
4.1 Oscillation of a Function 38
4.2 Null Selts 39
4.3 The Cantor Set and the Cantor Function 41
4.4 The Characterization for Continuous Integrators 42
V
vi Contents
4.5 The Characterization for General Integrators 46
Exercises 50
Chapter 5 Continuous Linear Functional on C[a, b] 54
5.1 The Norm on C [a, ft] 54
5.2 Positive Linear Functionals on C [a, ft] 56
5.3 Continuous Linear Functionals on C _a, ft] 60
5.4 Variation of a Function 64
5.5 Functions of Bounded Variation 66
Exercises 71
Chapter 6 Riemann Stieltjes Integrals: Further Properties 76
6.1 Integration by Parts 76
6.2 Fundamental Theorem of Calculus 79
6.3 A Theorem About Continuous Integrators 81
6.4 A Proof of Arzela s Theorem 82
Exercises 86
PART THREE LEBESGUE STIELTJES INTEGRALS 87
Chapter 7 The Extension of the Riemann Stieltjes Integral 89
7.1 The Extended Real Numbers 89
7.2 The Space CC(U) and Riemann Stieltjes Integrals 91
7.3 The First Extension of the Riemann Stieltjes Integral 91
7.4 Two Examples 96
Exercises 98
Chapter 8 Lebesgue Stieltjes Integrals 100
8.1 Lebesgue Stieltjes Integrals and Summable Functions 100
8.2 Two Examples 102
8.3 Linearity and Lattice Operations 105
8.4 Convergence Theorems 108
8.5 Riemann Stieltjes and Lebesgue Stieltjes Integrals 113
Exercises 117
PART FOUR MEASURE THEORY 121
Chapter 9 a Algebras and Algebras of Sets 123
9.1 a Algebras of Sets 123
9.2 The Borel a Algebra 124
Contents vii
9.3 Rings and Algebras of Sets 125
Exercises 126
Chapter 10 Measurable Functions 129
10.1 Simple Functions 129
10.2 Definition and Examples of Measurable Functions 130
10.3 Properties of Measurable Functions 132
10.4 Approximation by Simple Functions 135
Exercises 135
Chapter 11 Measures 137
11.1 Definitions and Examples of Measures 137
11.2 Measures in Probability Theory 138
11.3 Elementary Properties of Measures 139
11.4 Null Sets and Almost Everywhere 140
11.5 cr Finite and Semifinite Measures 141
11.6 Completion of a Measure 143
11.7 Outer Measures 145
11.8 Measures on Rings and Algebras of Sets 149
11.9 Atoms and Nonatomic Measures 153
Exercises 156
Chapter 12 Lebesgue Stieltjes Measures 160
12.1 The a Algebras 160
12.2 The Measures 162
12.3 Measurability of Summable Functions 163
12.4 The Integral in Terms of the Measure 164
12.5 Translation Invariance 165
12.6 The Role of Null Sets 168
12.7 Regularity of Lebesgue Stieltjes Measures 170
12.8 Characterization of Null Sets 171
12.9 Existence of Non Borel Sets of the Real Line 173
12.10 Lusin s Theorem 173
12.11 Characterizations of Lebesgue Stieltjes Measures 175
Exercises 180
PARTFIVE THE ABSTRACT LEBESGUE INTEGRAL 187
Chapter 13 The Integral Associated with a Measure Space 189
13.1 The Space of Simple Functions 189
13.2 Definition of the Abstract Lebesgue Integral 192
13.3 Properties of the Abstract Lebesgue Integral 194
viii Contents
13.4 Lebesgue Stieltjes Measures 202
13.5 Counting Measure 203
13.6 A Pathological Example 204
13.7 Complex Valued Functions 204
Exercises 206
Chapter 14 The Lebesgue Spaces and Norms 209
14.1 Pre Lebesgue Space and Minkowski s Inequality 209
14.2 Some Examples 214
14.3 Definition of the Lebesgue Spaces 217
14.4 Completeness of the Norm 220
14.5 Holder s Inequality 223
14.6 Applications of Holder s Inequality 226
14.7 Density Theorems 227
14.8 Inclusion Relations Amongst the Lebesgue Spaces 230
14.9 Jensen s Inequality 236
14.10 Convexity and Continuity of the Norm 240
Exercises 242
Chapter 15 Absolutely Continuous Measures 247
15.1 The Product of a Function and a Measure 247
15.2 Absolutely Continuous Measures 250
15.3 A Proof of the Radon Nikodym Theorem 253
15.4 Conditional Expectations 258
15.5 The Lebesgue Decomposition Theorem 262
Exercises 264
Chapter 16 Linear Functionals on the Lebesgue Spaces 268
16.1 Definition of the Canonical Maps 268
16.2 A Condition for the Canonical Maps to Be Isometric 270
16.3 A Condition for the Canonical Maps to Be Onto:
The Finite Case 271
16.4 A Condition for the Canonical Maps to Be Onto:
The General Case 275
16.5 A Characterization of L (X, Jt, i) 279
Exercises 280
Chapter 17 Product Measures and Fubini s Theorem 282
17.1 Iterated Integrals 282
17.2 Product ff Algebras 283
17.3 Product Measures 287
17.4 Integrals of Characteristic Functions 289
17.5 Fubini s and Tonelli s Theorems 293
Contents ix
17.6 Three Counterexamples 295
17.7 Products of More than Two Measure Spaces 297
Exercises 298
Chapter 18 Lebesgue Integration and Measure on W 302
18.1 Lebesgue Measure on K 302
18.2 An Analogue of Spherical Coordinates 307
18.3 Convolution Products 312
Exercises 314
Chapter 19 Signed Measures and Complex Measures 317
19.1 Definitions and Examples 317
19.2 Elementary Properties 320
19.3 Jordan and Hahn Decompositions 322
19.4 Existence of the Jordan Decomposition 323
19.5 Total Variation of Signed and Complex Measures 326
19.6 Norms of Real and Complex Measures 330
19.7 Lattice Operation on Signed Measures 332
19.8 Absolute Continuity and the Radon Nikodym Theorem 337
Exercises 338
Chapter 20 Differentiation 341
20.1 Vitali s Covering Theorem 341
20.2 Differentiation of Monotone Functions 344
20.3 Differentiation of Indefinite Integrals 348
20.4 Lebesgue Points and Points of Density 350
20.5 Absolutely Continuous Functions 353
20.6 Application to Lebesgue Stieltjes Measures 356
20.7 Differentiation of Functions on W 360
Exercises 364
Chapter 21 Convergence of Sequences of Functions 367
21.1 Definitions 367
21.2 Convergence on a General Measure Space 368
21.3 Convergence on a Finite Measure Space 374
21.4 Dominated Convergence 375
21.5 Summary 377
Exercises 378
Chapter 22 Measures on Locally Compact Spaces 381
22.1 Locally Compact Spaces 381
22.2 Regular Measures 383
22.3 Linear Functionals on Cc (X) and Cx (X) 386
x Contents
22.4 Completion of the Proof of Theorem 22.8 391
Exercises 396
Chapter 23 Hausdorff Measures and Dimension 400
23.1 A Property of Outer Measures 400
23.2 Hausdorff Measures 402
23.3 A Proof of Theorem 23.9 406
23.4 Hausdorff Dimension 412
Exercises 415
Chapter 24 Lorentz Spaces 417
24.1 Distribution Functions and Nonincreasing Rearrangements 417
24.2 The Lorentz Spaces 423
24.3 Norms on the Lorentz Spaces 426
24.4 Subadditivity of the Maximal Operator 433
Exercises 437
PART SIX APPENDICES 441
Appendix A Continuous Functions, Topology, and Set Theory 443
A.I Limit Superior and Limit Inferior of Sequences of Sets 443
A.2 Inverse Images 444
A.3 Inequalities Between and Operations on Functions 444
A.4 Elementary Topology 445
A.5 Pointwise and Uniform Convergence 446
A.6 Bounded and Continuous Functions 446
A.7 Axiom of Choice 449
A.8 Ordinal Numbers 450
A.9 Equivalence of Sets and Cardinal Numbers 451
Appendix B Functional Analysis 454
B.I Metric Spaces 454
B.2 Baire s Theorem 455
B.3 Norms 458
B.4 Continuous Linear Functionals 459
B.5 Inner Products 461
Index of Notation 465
Index 468
|
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| institution | BVB |
| isbn | 0471595187 |
| language | English |
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| series2 | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts A Wiley-interscience publication |
| spelling | Nielsen, Ole A. Verfasser aut An introduction to integration and measure theory Ole A. Nielsen New York [u.a.] Wiley 1997 XIV, 473 S. txt rdacontent n rdamedia nc rdacarrier Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts [16] A Wiley-interscience publication ANÁLISE ESPECTRAL (ANÁLISE FUNCIONAL) larpcal Integralen gtt Lebesgue-integralen gtt Maattheorie gtt Riemann-integralen gtt Integrals, Generalized Measure theory Integrationstheorie (DE-588)4138369-2 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 s DE-604 Maßtheorie (DE-588)4074626-4 s Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts [16] (DE-604)BV012067942 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007793344&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nielsen, Ole A. An introduction to integration and measure theory Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts ANÁLISE ESPECTRAL (ANÁLISE FUNCIONAL) larpcal Integralen gtt Lebesgue-integralen gtt Maattheorie gtt Riemann-integralen gtt Integrals, Generalized Measure theory Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4138369-2 (DE-588)4074626-4 |
| title | An introduction to integration and measure theory |
| title_auth | An introduction to integration and measure theory |
| title_exact_search | An introduction to integration and measure theory |
| title_full | An introduction to integration and measure theory Ole A. Nielsen |
| title_fullStr | An introduction to integration and measure theory Ole A. Nielsen |
| title_full_unstemmed | An introduction to integration and measure theory Ole A. Nielsen |
| title_short | An introduction to integration and measure theory |
| title_sort | an introduction to integration and measure theory |
| topic | ANÁLISE ESPECTRAL (ANÁLISE FUNCIONAL) larpcal Integralen gtt Lebesgue-integralen gtt Maattheorie gtt Riemann-integralen gtt Integrals, Generalized Measure theory Integrationstheorie (DE-588)4138369-2 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | ANÁLISE ESPECTRAL (ANÁLISE FUNCIONAL) Integralen Lebesgue-integralen Maattheorie Riemann-integralen Integrals, Generalized Measure theory Integrationstheorie Maßtheorie |
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