Measure theory: 1
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
[2007]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 500 Seiten |
| ISBN: | 9783540345138 |
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MARC
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| 100 | 1 | |a Bogačev, Vladimir I. |d 1961- |e Verfasser |0 (DE-588)121192318 |4 aut | |
| 245 | 1 | 0 | |a Measure theory |n 1 |c V. I. Bogachev |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c [2007] | |
| 300 | |a xvii, 500 Seiten | ||
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Record in the Search Index
| _version_ | 1804136458091495424 |
|---|---|
| adam_text | Contents
Preface
..................................................................v
Chapter
1.
Constructions and extensions of measures
............1
1.1.
Measurement of length: introductory remarks
.................1
1.2.
Algebras and
σ
-algebras
.......................................
3
1.3.
Additivity and countable additivity of measures
...............9
1.4.
Compact classes and countable additivity
....................13
1.5.
Outer measure and the Lebesgue extension of measures
.......16
1.6.
Infinite and
σ
-nnite
measures
................................24
1.7.
Lebesgue measure
............................................26
1.8.
Lebesgue-
Stieltjes
measures
..................................32
1.9.
Monotone and
σ
-additive
classes of sets
......................33
1.10.
Souslin sets and the A-operation
.............................35
1.11.
Caratheodory outer measures
................................41
1.12.
Supplements and exercises
...................................48
Set operations
(48).
Compact classes
(50).
Metric Boolean algebra
(53).
Measurable envelope, measurable kernel and inner measure
(56).
Extensions of measures
(58).
Some interesting sets
(61).
Additive, but
not countably additive measures
(67).
Abstract inner measures
(70).
Measures on lattices of sets
(75).
Set-theoretic problems in measure
theory
(77).
Invariant extensions of Lebesgue measure
(80).
Whitney s
decomposition
(82).
Exercises
(83).
Chapter
2.
The Lebesgue integral
................................105
2.1.
Measurable functions
.......................................105
2.2.
Convergence in measure and almost everywhere
.............110
2.3.
The integral for simple functions
............................115
2.4.
The general definition of the Lebesgue integral
..............118
2.5.
Basic properties of the integral
..............................121
2.6.
Integration with respect to infinite measures
................124
2.7.
The completeness of the space L1
...........................128
2.8.
Convergence theorems
......................................130
2.9.
Criteria of integrability
.....................................136
2.10.
Connections with the Riemann integral
.....................138
2.11.
The Holder and Minkowski inequalities
......................139
2.12. Supplements
and exercises
..................................143
The
σ
-algebra
generated by a class of functions
(143).
Borei
mappings
on IRn
(145).
The functional monotone class theorem
(146).
Baire
classes of functions
(148).
Mean value theorems
(150).
The Lebesgue
Stieltjes
integral
(152).
Integral inequalities
(153).
Exercises
(156).
Chapter
3.
Operations on measures and functions
..............175
3.1.
Decomposition of signed measures
...........................175
3.2.
The Radon Nikodym theorem
..............................177
3.3.
Products of measure spaces
.................................180
3.4.
Fubini s theorem
............................................183
3.5.
Infinite products of measures
................................187
3.6.
Images of measures under mappings
.........................190
3.7.
Change of variables in IRn
..................................194
3.8.
The Fourier transform
......................................197
3.9.
Convolution
.................................................204
3.10.
Supplements and exercises
..................................209
On Fubini s theorem and products of
σ
-algebras
(209).
Steiner s
symmetrization
(212).
Hausdorff measures
(215).
Decompositions of
set functions
(218).
Properties of positive definite functions
(220).
The Brunn-Minkowski inequality and its generalizations
(222),
Mixed volumes
(226).
Exercises
(228).
Chapter
4.
The spaces Lp and spaces of measures
..............249
4.1.
The spaces Lp
..............................................249
4.2.
Approximations in LP
.......................................251
4.3.
The Hubert space L2
.......................................254
4.4.
Duality of the spaces Lp
....................................262
4.5.
Uniform integrability
........................................266
4.6.
Convergence of measures
....................................273
4.7.
Supplements and exercises
..................................277
The spaces Lp and the space of measures as structures
(277).
The weak
topology in Lp
(280).
Uniform convexity
(283).
Uniform integrability
and weak compactness in L1
(285).
The topology of setwise convergence
of measures
(291).
Norm compactness and approximations in Lp
(294).
Certain conditions of convergence in Lp
(298).
Hellinger s integral and
Hellinger
s
distance
(299).
Additive set functions
(302).
Exercises
(303).
Chapter
5.
Connections between the integral and derivative
.. 329
5.1.
Differentiability of functions on the real line
.................329
5.2.
Functions of bounded variation
..............................332
5.3.
Absolutely continuous functions
.............................337
5.4.
The Newton-Leibniz formula
................................341
5.5.
Covering theorems
..........................................345
5.6.
The maximal function
.......................................349
5.7.
The Henstock-Kurzweil integral
.............................353
5.8. Supplements
and exercises
..................................361
Covering theorems
(361).
Density points and Lebesgue points
(366).
Differentiation of measures on IRn
(367).
The approximate
continuity
(369). Derivates
and the approximate differentiability
(370).
The class BMO
(373).
Weighted inequalities
(374).
Measures with
the doubling property
(375).
Sobolev derivatives
(376).
The area and
coarea
formulas and change of variables
(379).
Surface measures
(383).
The
Calderón-Zygmund
decomposition
(385).
Exercises
(386).
Bibliographical and Historical Comments
.........................409
References
............................................................441
Author Index
........................................................483
Subject Index
........................................................491
|
| adam_txt |
Contents
Preface
.v
Chapter
1.
Constructions and extensions of measures
.1
1.1.
Measurement of length: introductory remarks
.1
1.2.
Algebras and
σ
-algebras
.
3
1.3.
Additivity and countable additivity of measures
.9
1.4.
Compact classes and countable additivity
.13
1.5.
Outer measure and the Lebesgue extension of measures
.16
1.6.
Infinite and
σ
-nnite
measures
.24
1.7.
Lebesgue measure
.26
1.8.
Lebesgue-
Stieltjes
measures
.32
1.9.
Monotone and
σ
-additive
classes of sets
.33
1.10.
Souslin sets and the A-operation
.35
1.11.
Caratheodory outer measures
.41
1.12.
Supplements and exercises
.48
Set operations
(48).
Compact classes
(50).
Metric Boolean algebra
(53).
Measurable envelope, measurable kernel and inner measure
(56).
Extensions of measures
(58).
Some interesting sets
(61).
Additive, but
not countably additive measures
(67).
Abstract inner measures
(70).
Measures on lattices of sets
(75).
Set-theoretic problems in measure
theory
(77).
Invariant extensions of Lebesgue measure
(80).
Whitney's
decomposition
(82).
Exercises
(83).
Chapter
2.
The Lebesgue integral
.105
2.1.
Measurable functions
.105
2.2.
Convergence in measure and almost everywhere
.110
2.3.
The integral for simple functions
.115
2.4.
The general definition of the Lebesgue integral
.118
2.5.
Basic properties of the integral
.121
2.6.
Integration with respect to infinite measures
.124
2.7.
The completeness of the space L1
.128
2.8.
Convergence theorems
.130
2.9.
Criteria of integrability
.136
2.10.
Connections with the Riemann integral
.138
2.11.
The Holder and Minkowski inequalities
.139
2.12. Supplements
and exercises
.143
The
σ
-algebra
generated by a class of functions
(143).
Borei
mappings
on IRn
(145).
The functional monotone class theorem
(146).
Baire
classes of functions
(148).
Mean value theorems
(150).
The Lebesgue
Stieltjes
integral
(152).
Integral inequalities
(153).
Exercises
(156).
Chapter
3.
Operations on measures and functions
.175
3.1.
Decomposition of signed measures
.175
3.2.
The Radon Nikodym theorem
.177
3.3.
Products of measure spaces
.180
3.4.
Fubini's theorem
.183
3.5.
Infinite products of measures
.187
3.6.
Images of measures under mappings
.190
3.7.
Change of variables in IRn
.194
3.8.
The Fourier transform
.197
3.9.
Convolution
.204
3.10.
Supplements and exercises
.209
On Fubini's theorem and products of
σ
-algebras
(209).
Steiner's
symmetrization
(212).
Hausdorff measures
(215).
Decompositions of
set functions
(218).
Properties of positive definite functions
(220).
The Brunn-Minkowski inequality and its generalizations
(222),
Mixed volumes
(226).
Exercises
(228).
Chapter
4.
The spaces Lp and spaces of measures
.249
4.1.
The spaces Lp
.249
4.2.
Approximations in LP
.251
4.3.
The Hubert space L2
.254
4.4.
Duality of the spaces Lp
.262
4.5.
Uniform integrability
.266
4.6.
Convergence of measures
.273
4.7.
Supplements and exercises
.277
The spaces Lp and the space of measures as structures
(277).
The weak
topology in Lp
(280).
Uniform convexity
(283).
Uniform integrability
and weak compactness in L1
(285).
The topology of setwise convergence
of measures
(291).
Norm compactness and approximations in Lp
(294).
Certain conditions of convergence in Lp
(298).
Hellinger's integral and
Hellinger
's
distance
(299).
Additive set functions
(302).
Exercises
(303).
Chapter
5.
Connections between the integral and derivative
. 329
5.1.
Differentiability of functions on the real line
.329
5.2.
Functions of bounded variation
.332
5.3.
Absolutely continuous functions
.337
5.4.
The Newton-Leibniz formula
.341
5.5.
Covering theorems
.345
5.6.
The maximal function
.349
5.7.
The Henstock-Kurzweil integral
.353
5.8. Supplements
and exercises
.361
Covering theorems
(361).
Density points and Lebesgue points
(366).
Differentiation of measures on IRn
(367).
The approximate
continuity
(369). Derivates
and the approximate differentiability
(370).
The class BMO
(373).
Weighted inequalities
(374).
Measures with
the doubling property
(375).
Sobolev derivatives
(376).
The area and
coarea
formulas and change of variables
(379).
Surface measures
(383).
The
Calderón-Zygmund
decomposition
(385).
Exercises
(386).
Bibliographical and Historical Comments
.409
References
.441
Author Index
.483
Subject Index
.491 |
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| id | DE-604.BV022397611 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:17:05Z |
| indexdate | 2024-07-09T20:56:43Z |
| institution | BVB |
| isbn | 9783540345138 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015606325 |
| oclc_num | 315569787 |
| open_access_boolean | |
| owner | DE-824 DE-703 DE-573 DE-29T DE-634 DE-20 DE-83 DE-188 DE-706 DE-739 DE-19 DE-BY-UBM |
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| physical | xvii, 500 Seiten |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| spelling | Bogačev, Vladimir I. 1961- Verfasser (DE-588)121192318 aut Measure theory 1 V. I. Bogachev Berlin [u.a.] Springer [2007] xvii, 500 Seiten txt rdacontent n rdamedia nc rdacarrier Maßtheorie (DE-588)4074626-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 s DE-604 (DE-604)BV022397610 1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015606325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bogačev, Vladimir I. 1961- Measure theory Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4074626-4 |
| title | Measure theory |
| title_auth | Measure theory |
| title_exact_search | Measure theory |
| title_exact_search_txtP | Measure theory |
| title_full | Measure theory 1 V. I. Bogachev |
| title_fullStr | Measure theory 1 V. I. Bogachev |
| title_full_unstemmed | Measure theory 1 V. I. Bogachev |
| title_short | Measure theory |
| title_sort | measure theory |
| topic | Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015606325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022397610 |
| work_keys_str_mv | AT bogacevvladimiri measuretheory1 |