Real analysis: theory of measure and integration
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2008
|
| Ausgabe: | 2. ed., reprint. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 1. Aufl. u.d.T.: Yeh, James: Lectures on real analysis Includes bibliographical references (p. 727-728) and index |
| Beschreibung: | XXI, 738 S. 23 cm |
| ISBN: | 9789812566546 9812566546 9812566538 9789812566539 |
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| 245 | 1 | 0 | |a Real analysis |b theory of measure and integration |c J. Yeh |
| 250 | |a 2. ed., reprint. | ||
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2008 | |
| 300 | |a XXI, 738 S. |c 23 cm | ||
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| 500 | |a 1. Aufl. u.d.T.: Yeh, James: Lectures on real analysis | ||
| 500 | |a Includes bibliographical references (p. 727-728) and index | ||
| 650 | 4 | |a Measure theory | |
| 650 | 4 | |a Lebesgue integral | |
| 650 | 4 | |a Integrals, Generalized | |
| 650 | 4 | |a Mathematical analysis | |
| 650 | 4 | |a Lp spaces | |
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Datensatz im Suchindex
| _version_ | 1804140734837686272 |
|---|---|
| adam_text | Titel: Real analysis
Autor: Yeh, James
Jahr: 2008
Contents
Preface to the First Edition xiii
Preface to the Second Edition xvii
Notations xix
1 Measure Spaces 1
§0 Introduction.................................. 1
§1 Measure on a a -algebra of Sets........................ 3
[I] (T-algebra of Sets............................ 3
[II] Limits of Sequences of Sets...................... 4
[III] Generation of a -algebras ....................... 6
[IV] Borel a -algebras............................ 9
[V] Measure on a a -algebra........................ 11
[VI] Measures of a Sequence of Sets.................... 14
[VII] Measurable Space and Measure Space ................ 17
[VIII] Measurable Mapping.......................... 19
[IX] Induction of Measure by Measurable Mapping............ 22
§2 Outer Measures................................ 28
[I] Construction of Measure by Means of Outer Measure........ 28
[II] Regular Outer Measures........................ 32
[III] Metric Outer Measures ........................ 34
[IV] Construction of Outer Measures.................... 37
§3 Lebesgue Measure on K ........................... 41
[I] Lebesgue Outer Measure on R..................... 41
[II] Some Properties of the Lebesgue Measure Space........... 45
[III] Existence of Non Lebesgue Measurable Sets............. 49
[IV] Regularity of Lebesgue Outer Measure................ 51
[V] Lebesgue Inner Measure on R..................... 57
§4 Measurable Functions ............................ 70
[I] Measurability of Functions ...................... 70
[II] Operations with Measurable Functions................ 74
[III] Equality Almost Everywhere..................... 78
[IV] Sequence of Measurable Functions.................. 79
[V] Continuity and Borel and Lebesgue Measurability of Functions on E 83
vii
Vln Contents
[VI] Cantor Ternary Set and Cantor-Lebesgue Function..........85
§5 Completion of Measure Space........................95
[I] Complete Extension and Completion of a Measure Space......95
[II] Completion of the Borel Measure Space to the Lebesgue Measure
Space..................................98
§6 Convergence a.e. and Convergence in Measure ...............100
[I] Convergence a.e.............................100
[IT] Almost Uniform Convergence.....................104
[III] Convergence in Measure .......................107
[IV] Cauchy Sequences in Convergence in Measure............112
[V] Approximation by Step Functions and Continuous Functions .... 115
2 The Lebesgue Integral 127
§7 Integration of Bounded Functions on Sets of Finite Measure........127
[I] Integration of Simple Functions....................127
[II] Integration of Bounded Functions on Sets of Finite Measure.....131
[III] Riemann Integrability.........................140
§8 Integration of Nonnegative Functions ....................152
[I] Lebesgue Integral of Nonnegative Functions.............152
[II] Monotone Convergence Theorem...................154
[IH] Approximation of the Integral by Truncation.............162
§9 Integration of Measurable Functions.....................169
[I] Lebesgue Integral of Measurable Functions..............169
[II] Convergence Theorems ........................178
[III] Convergence Theorems under Convergence in Measure.......182
[IV] Approximation of the Integral by Truncation.............183
[V] Translation and Linear Transformation in the Lebesgue Integral on R 189
[VI] Integration by Image Measure ....................193
§10 Signed Measures...............................202
[I] Signed Measure Spaces........................202
[II] Decomposition of Signed Measures..................208
[III] Integration on a Signed Measure Space................217
§ 11 Absolute Continuity of a Measure......................224
[I] The Radon-Nikodym Derivative ...................224
[II] Absolute Continuity of a Signed Measure Relative to a Positive
Measure................................225
[III] Properties of the Radon-Nikodym Derivative.............236
3 Differentiation and Integration 245
§12 Monotone Functions and Functions of Bounded Variation..........245
[I] The Derivative.............................245
[II] Differentiability of Monotone Functions ...............251
[III] Functions of Bounded Variation....................261
§13 Absolutely Continuous Functions......................270
[I] Absolute Continuity..........................270
Contents ix
[II] Banach-Zarecki Criterion for Absolute Continuity..........273
[HI] Singular Functions ..........................276
[IV] Indefinite Integrals ..........................276
[V] Calculation of the Lebesgue Integral by Means of the Derivative . . 287
[VI] Length of Rectifiable Curves .....................298
§14 Convex Functions ..............................308
[I] Continuity and Differentiability of a Convex Function........308
[II] Monotonicity and Absolute Continuity of a Convex Function . . . .317
[ÝÜ] Jensen s Inequality...........................320
4 The Classical Banach Spaces 323
§15 Normed Linear Spaces............................323
[I] Banach Spaces.............................323
[II] Banach Spaces on R* .........................326
[III] The Space of Continuous Functions C([a, b]).............329
[IV] A Criterion for Completeness of a Normed Linear Space ......331
[V] Hilbert Spaces.............................333
[VI] Bounded Linear Mappings of Normed Linear Spaces.........334
[VII] Baire Category Theorem........................344
[Vffl] Uniform Boundedness Theorems...................347
[IX] Open Mapping Theorem........................350
[X] Hahn-Banach Extension Theorems..................357
[XI] Semicontinuous Functions.......................370
§16 The f Spaces ................................376
[I] The Lp Spaces for p e (0, oo) ....................376
[n] The Linear Spaces Lp for p 6 [l,oo).................379
[III] The LP Spaces forp 6 [1, oo).....................384
[IV] The Space L00.............................393
[V] The LP Spaces for p € (0,1).....................401
[VI] Extensions of Holder s Inequality...................406
§17 Relation among the Lp Spaces........................412
[I] The Modified LP Norms for Lp Spaces with p€ [l,co].......412
[II] Approximation by Continuous Functions.............. . 414
[III] LP Spaces with p e (0, 1].......................417
[IV] The IP Spaces.............................422
§18 Bounded Linear Functionals on the Lp Spaces...............429
[I] Bounded Linear Functionals Arising from Integration........429
[II] Approximation by Simple Functions .................432
[III] A Converse of Holder s Inequality...................434
[IV] Riesz Representation Theorem on the V Spaces...........437
§19 Integration on Locally Compact Hausdorff Space..............445
[I] Continuous Functions on a Locally Compact Hausdorff Space . . .445
[II] Borel and Radon Measures ......................450
[III] Positive Linear Functionals on CC(X).................455
[IV] Approximation by Continuous Functions...............463
x Contents
[V] Signed Radon Measures........................467
[VI] The Dual Space of C(X)........................471
5 Extension of Additive Set Functions to Measures 481
§20 Extension of Additive Set Functions on an Algebra.............481
[I] Additive Set Function on an Algebra.................481
[II] Extension of an Additive Set Function on an Algebra to a Measure . 486
[III] Regularity of an Outer Measure Derived from a Countably Additive
Set Function on an Algebra......................486
[IV] Uniqueness of Extension of a Countably Additive Set Function on
an Algebra to a Measure .......................489
[V] Approximation toa a -algebra Generated by an Algebra.......491
[VI] Outer Measure Based on a Measure..................494
§21 Extension of Additive Set Functions on a Semialgebra ...........496
[I] Semialgebras of Sets .........................496
[II] Additive Set Function on a Semialgebra ...............498
[III] Outer Measures Based on Additive Set Functions on a Semialgebra . 502
§22 Lebesgue-Stieltjes Measure Spaces.....................505
[I] Lebesgue-Stieltjes Outer Measures..................505
[II] Regularity of the Lebesgue-Stieltjes Outer Measures.........509
[III] Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure 511
[IV] Decomposition of an Increasing Function...............519
§23 Product Measure Spaces...........................527
[I] Existence and Uniqueness of Product Measure Spaces........527
[II] Integration on Product Measure Space ................531
[III] Completion of Product Measure Space................543
[IV] Convolution of Functions.......................547
[V] Some Related Theorems........................587
6 Measure and Integration on the Euclidean Space 597
§24 Lebesgue Measure Space on the Euclidean Space..............597
[I] Lebesgue Outer Measure on the Euclidean Space ..........597
[II] Regularity Properties of Lebesgue Measure Space on K .......602
[III] Approximation by Continuous Functions...............605
[IV] Lebesgue Measure Space on W as the Completion of a Product
Measure Space.............................609
[V] Translation of the Lebesgue Integral on M ..............610
[VI] Linear Transformation of the Lebesgue Integral on K ........612
§25 Differentiation on the Euclidean Space ...................620
[I] The Lebesgue Differentiation Theorem on R ............620
[II] Differentiation of Set Functions with Respect to the Lebesgue
Measure................................632
[III] Differentiation of the Indefinite Integral................634
[IV] Density of Lebesgue Measurable Sets Relative to the Lebesgue
Measure................................635
Contents xi
[V] Signed Borel Measures on R .....................641
[VI] Differentiation of Borel Measures with Respect to the Lebesgue
Measure................................643
§26 Change of Variable of Integration on the Euclidean Space .........649
[I] Change of Variable of Integration by Differentiable Transformations 649
[II] Spherical Coordinates in M ......................661
[IH] Integration by Image Measure on Spherical Surfaces.........667
7 Hausdorff Measures on the Euclidean Space 675
§27 Hausdorff Measures.............................675
[I] Hausdorff Measures on R .......................675
[II] Equivalent Definitions of Hausdorff Measure.............680
[III] Regularity of Hausdorff Measure...................686
[IV] Hausdorff Dimension .........................689
§28 Transformations of Hausdorff Measures...................694
[I] Hausdorff Measure of Transformed Sets ...............694
[II] 1-dimensional Hausdorff Measure ..................699
[III] Hausdorff Measure of Jordan Curves.................700
§29 Hausdorff Measures of Integral and Fractional Dimensions.........705
[I] Hausdorff Measure of Integral Dimension and Lebesgue Measure . . 705
[II] Calculation of the n-dimensional Hausdorff Measure of a Unit Cube
in R ..................................707
[III] Transformation of Hausdorff Measure of Integral Dimension.....713
[IV] Hausdorff Measure of Fractional Dimension.............718
Bibliography 727
Index 729
|
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| spelling | Yeh, James Verfasser aut Real analysis theory of measure and integration J. Yeh 2. ed., reprint. Singapore [u.a.] World Scientific 2008 XXI, 738 S. 23 cm txt rdacontent n rdamedia nc rdacarrier 1. Aufl. u.d.T.: Yeh, James: Lectures on real analysis Includes bibliographical references (p. 727-728) and index Measure theory Lebesgue integral Integrals, Generalized Mathematical analysis Lp spaces Maßtheorie (DE-588)4074626-4 gnd rswk-swf Reelle Analysis (DE-588)4627581-2 gnd rswk-swf Integrationstheorie (DE-588)4138369-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Reelle Analysis (DE-588)4627581-2 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=018652288&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yeh, James Real analysis theory of measure and integration Measure theory Lebesgue integral Integrals, Generalized Mathematical analysis Lp spaces Maßtheorie (DE-588)4074626-4 gnd Reelle Analysis (DE-588)4627581-2 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| subject_GND | (DE-588)4074626-4 (DE-588)4627581-2 (DE-588)4138369-2 (DE-588)4123623-3 |
| title | Real analysis theory of measure and integration |
| title_auth | Real analysis theory of measure and integration |
| title_exact_search | Real analysis theory of measure and integration |
| title_full | Real analysis theory of measure and integration J. Yeh |
| title_fullStr | Real analysis theory of measure and integration J. Yeh |
| title_full_unstemmed | Real analysis theory of measure and integration J. Yeh |
| title_short | Real analysis |
| title_sort | real analysis theory of measure and integration |
| title_sub | theory of measure and integration |
| topic | Measure theory Lebesgue integral Integrals, Generalized Mathematical analysis Lp spaces Maßtheorie (DE-588)4074626-4 gnd Reelle Analysis (DE-588)4627581-2 gnd Integrationstheorie (DE-588)4138369-2 gnd |
| topic_facet | Measure theory Lebesgue integral Integrals, Generalized Mathematical analysis Lp spaces Maßtheorie Reelle Analysis Integrationstheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018652288&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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