Real analysis and probability:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
2004
|
| Ausgabe: | 2. ed., repr. |
| Schriftenreihe: | Cambridge studies in advanced mathematics
74 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 555 S. |
| ISBN: | 052180972X 9780521809726 9780521007542 0521007542 |
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Datensatz im Suchindex
| _version_ | 1805073839724429312 |
|---|---|
| adam_text |
Contents
Preface
to the Cambridge Edition page
ix
1
Foundations; Set Theory
1
1.1
Definitions for Set Theory and the Real Number System
1
1.2
Relations and
Orderings
9
*1
.3
Transfinite
Induction and Recursion
12
1.4
Cardinality
16
1.5
The Axiom of Choice and Its Equivalents
18
2
General Topology
24
2.1
Topologies, Metrics, and Continuity
24
2.2
Compactness and Product Topologies
34
2.3
Complete and Compact Metric Spaces
44
2.4
Some Metrics for Function Spaces
48
2.5
Completion and Completeness of Metric Spaces
58
*2.6 Extension of Continuous Functions
63
*2.7 Uniformities and Uniform Spaces
67
*2.8 Compactification
71
3
Measures
85
3.1
Introduction to Measures
85
3.2
Semirings and Rings
94
3.3
Completion of Measures
101
3.4
Lebesgue Measure and Nonmeasurable Sets
105
*3.5 Atomic and Nonatomic Measures
109
4
Integration
114
4.1
Simple Functions
114
*4.2 Measurability
123
4.3
Convergence Theorems for Integrals
130
vi
Contents
4.4
Product
Measures
134
*4.5 Daniell-Stone
Integrals
142
5
Lp
Spaces; Introduction to Functional Analysis
152
5.1
Inequalities for Integrals
152
5.2
Norms and Completeness of LP
158
5.3
Hubert Spaces
160
5.4
Orthonormal
Sets and Bases
165
5.5
Linear Forms on Hubert Spaces, Inclusions of LP Spaces,
and Relations Between Two Measures
173
5.6
Signed Measures
178
6
Convex Sets and Duality of Normed Spaces
188
6.1
Lipschitz, Continuous, and Bounded Functionals
188
6.2
Convex Sets and Their Separation
195
6.3
Convex Functions
203
*6.4 Duality of LP Spaces
208
6.5
Uniform Boundedness and Closed Graphs
211
*6.6 The Brunn-Minkowski Inequality
215
7
Measure, Topology, and Differentiation
222
7.1
Baire and
Borei
σ-
Algebras and Regularity of Measures
222
*7.2 Lebesgue's Differentiation Theorems
228
*7.3 The Regularity Extension
235
*7.4 The Dual of C{K) and Fourier Series
239
*7.5 Almost Uniform Convergence and Lusin's Theorem
243
8
Introduction to Probability Theory
250
8.1
Basic Definitions
251
8.2
Infinite Products of Probability Spaces
255
8.3
Laws of Large Numbers
260
*8.4 Ergodic Theorems
267
9
Convergence of Laws and Central Limit Theorems
282
9.1
Distribution Functions and Densities
282
9.2
Convergence of Random Variables
287
9.3
Convergence of Laws
291
9.4
Characteristic Functions
298
9.5
Uniqueness of Characteristic Functions
and a Central Limit Theorem
303
9.6
Triangular Arrays and Lindeberg's Theorem
315
9.7
Sums of Independent Real Random Variables
320
Contents
vii
*9.8
The Levy Continuity Theorem; Infinitely Divisible
and Stable Laws
325
10
Conditional Expectations and Martingales
336
10.1
Conditional Expectations
336
10.2
Regular Conditional Probabilities and Jensen's
Inequality
341
10.3
Martingales
353
10.4
Optional Stopping and Uniform Integrability
358
10.5
Convergence of Martingales and
Submartingales
364
*10.6 Reversed Martingales and
Submartingales
370
* 10.7 Subadditive
and
Superadditive
Ergodic Theorems
374
11
Convergence of Laws on Separable Metric Spaces
385
11.1
Laws and Their Convergence
385
11.2
Lipschitz Functions
390
11.3
Metrics for Convergence of Laws
393
11.4
Convergence of Empirical Measures
399
11.5
Tightness and Uniform Tightness
402
*1
1.6
Strassen's Theorem: Nearby Variables
with Nearby Laws
406
* 11.7
A Uniformity for Laws and Almost Surely Converging
Realizations of Converging Laws
413
*1
1.8
Kantorovich-Rubinstein Theorems
420
*11.9 ^-Statistics
426
12
Stochastic Processes
439
12.1
Existence of Processes and Brownian Motion
439
12.2
The Strong Markov Property of Brownian Motion
450
12.3
Reflection Principles, The Brownian Bridge,
and Laws of
Suprema
459
12.4
Laws of Brownian Motion at Markov Times:
Skorohod Imbedding
469
12.5
Laws of the Iterated Logarithm
476
13
Measurability:
Borei
Isomorphism and Analytic Sets
487
*13.1
Borei
Isomorphism
487
*13.2 Analytic Sets
493
Appendix A Axiomatic Set Theory
503
A.
1
Mathematical Logic
503
A.2 Axioms for Set Theory
505
viii Contents
А.З
Ordinals and Cardinals
510
A.4 From Sets to Numbers
515
Appendix
В
Complex Numbers, Vector Spaces,
and Taylor's Theorem with Remainder
521
Appendix
С
The Problem of Measure
526
Appendix
D
Rearranging Sums of
Nonnegative
Terms
528
Appendix
E
Pathologies of Compact Nonmetric Spaces
530
Author Index
541
Subject Index
546
Notation Index
554 |
| adam_txt |
Contents
Preface
to the Cambridge Edition page
ix
1
Foundations; Set Theory
1
1.1
Definitions for Set Theory and the Real Number System
1
1.2
Relations and
Orderings
9
*1
.3
Transfinite
Induction and Recursion
12
1.4
Cardinality
16
1.5
The Axiom of Choice and Its Equivalents
18
2
General Topology
24
2.1
Topologies, Metrics, and Continuity
24
2.2
Compactness and Product Topologies
34
2.3
Complete and Compact Metric Spaces
44
2.4
Some Metrics for Function Spaces
48
2.5
Completion and Completeness of Metric Spaces
58
*2.6 Extension of Continuous Functions
63
*2.7 Uniformities and Uniform Spaces
67
*2.8 Compactification
71
3
Measures
85
3.1
Introduction to Measures
85
3.2
Semirings and Rings
94
3.3
Completion of Measures
101
3.4
Lebesgue Measure and Nonmeasurable Sets
105
*3.5 Atomic and Nonatomic Measures
109
4
Integration
114
4.1
Simple Functions
114
*4.2 Measurability
123
4.3
Convergence Theorems for Integrals
130
vi
Contents
4.4
Product
Measures
134
*4.5 Daniell-Stone
Integrals
142
5
Lp
Spaces; Introduction to Functional Analysis
152
5.1
Inequalities for Integrals
152
5.2
Norms and Completeness of LP
158
5.3
Hubert Spaces
160
5.4
Orthonormal
Sets and Bases
165
5.5
Linear Forms on Hubert Spaces, Inclusions of LP Spaces,
and Relations Between Two Measures
173
5.6
Signed Measures
178
6
Convex Sets and Duality of Normed Spaces
188
6.1
Lipschitz, Continuous, and Bounded Functionals
188
6.2
Convex Sets and Their Separation
195
6.3
Convex Functions
203
*6.4 Duality of LP Spaces
208
6.5
Uniform Boundedness and Closed Graphs
211
*6.6 The Brunn-Minkowski Inequality
215
7
Measure, Topology, and Differentiation
222
7.1
Baire and
Borei
σ-
Algebras and Regularity of Measures
222
*7.2 Lebesgue's Differentiation Theorems
228
*7.3 The Regularity Extension
235
*7.4 The Dual of C{K) and Fourier Series
239
*7.5 Almost Uniform Convergence and Lusin's Theorem
243
8
Introduction to Probability Theory
250
8.1
Basic Definitions
251
8.2
Infinite Products of Probability Spaces
255
8.3
Laws of Large Numbers
260
*8.4 Ergodic Theorems
267
9
Convergence of Laws and Central Limit Theorems
282
9.1
Distribution Functions and Densities
282
9.2
Convergence of Random Variables
287
9.3
Convergence of Laws
291
9.4
Characteristic Functions
298
9.5
Uniqueness of Characteristic Functions
and a Central Limit Theorem
303
9.6
Triangular Arrays and Lindeberg's Theorem
315
9.7
Sums of Independent Real Random Variables
320
Contents
vii
*9.8
The Levy Continuity Theorem; Infinitely Divisible
and Stable Laws
325
10
Conditional Expectations and Martingales
336
10.1
Conditional Expectations
336
10.2
Regular Conditional Probabilities and Jensen's
Inequality
341
10.3
Martingales
353
10.4
Optional Stopping and Uniform Integrability
358
10.5
Convergence of Martingales and
Submartingales
364
*10.6 Reversed Martingales and
Submartingales
370
* 10.7 Subadditive
and
Superadditive
Ergodic Theorems
374
11
Convergence of Laws on Separable Metric Spaces
385
11.1
Laws and Their Convergence
385
11.2
Lipschitz Functions
390
11.3
Metrics for Convergence of Laws
393
11.4
Convergence of Empirical Measures
399
11.5
Tightness and Uniform Tightness
402
*1
1.6
Strassen's Theorem: Nearby Variables
with Nearby Laws
406
* 11.7
A Uniformity for Laws and Almost Surely Converging
Realizations of Converging Laws
413
*1
1.8
Kantorovich-Rubinstein Theorems
420
*11.9 ^-Statistics
426
12
Stochastic Processes
439
12.1
Existence of Processes and Brownian Motion
439
12.2
The Strong Markov Property of Brownian Motion
450
12.3
Reflection Principles, The Brownian Bridge,
and Laws of
Suprema
459
12.4
Laws of Brownian Motion at Markov Times:
Skorohod Imbedding
469
12.5
Laws of the Iterated Logarithm
476
13
Measurability:
Borei
Isomorphism and Analytic Sets
487
*13.1
Borei
Isomorphism
487
*13.2 Analytic Sets
493
Appendix A Axiomatic Set Theory
503
A.
1
Mathematical Logic
503
A.2 Axioms for Set Theory
505
viii Contents
А.З
Ordinals and Cardinals
510
A.4 From Sets to Numbers
515
Appendix
В
Complex Numbers, Vector Spaces,
and Taylor's Theorem with Remainder
521
Appendix
С
The Problem of Measure
526
Appendix
D
Rearranging Sums of
Nonnegative
Terms
528
Appendix
E
Pathologies of Compact Nonmetric Spaces
530
Author Index
541
Subject Index
546
Notation Index
554 |
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| id | DE-604.BV035118006 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:20:20Z |
| indexdate | 2024-07-20T05:15:59Z |
| institution | BVB |
| isbn | 052180972X 9780521809726 9780521007542 0521007542 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016785722 |
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| open_access_boolean | |
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| owner_facet | DE-739 DE-19 DE-BY-UBM |
| physical | X, 555 S. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Dudley, Richard M. 1938- Verfasser (DE-588)121010996 aut Real analysis and probability R. M. Dudley 2. ed., repr. Cambridge Cambridge Univ. Press 2004 X, 555 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 74 Functions of real variables Mathematical analysis Probabilities Reelle Funktion (DE-588)4048918-8 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Analysis (DE-588)4001865-9 s Reelle Funktion (DE-588)4048918-8 s Cambridge studies in advanced mathematics 74 (DE-604)BV000003678 74 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016785722&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dudley, Richard M. 1938- Real analysis and probability Cambridge studies in advanced mathematics Functions of real variables Mathematical analysis Probabilities Reelle Funktion (DE-588)4048918-8 gnd Analysis (DE-588)4001865-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4048918-8 (DE-588)4001865-9 (DE-588)4064324-4 |
| title | Real analysis and probability |
| title_auth | Real analysis and probability |
| title_exact_search | Real analysis and probability |
| title_exact_search_txtP | Real analysis and probability |
| title_full | Real analysis and probability R. M. Dudley |
| title_fullStr | Real analysis and probability R. M. Dudley |
| title_full_unstemmed | Real analysis and probability R. M. Dudley |
| title_short | Real analysis and probability |
| title_sort | real analysis and probability |
| topic | Functions of real variables Mathematical analysis Probabilities Reelle Funktion (DE-588)4048918-8 gnd Analysis (DE-588)4001865-9 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Functions of real variables Mathematical analysis Probabilities Reelle Funktion Analysis Wahrscheinlichkeitsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016785722&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT dudleyrichardm realanalysisandprobability |