Generalized measure theory:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Series: | IFSR international series on systems science and engineering
25 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XV, 381 S. Ill., graph. Darst. |
| ISBN: | 0387768513 9780387768519 |
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MARC
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| 245 | 1 | 0 | |a Generalized measure theory |c Zhenyuan Wang ; George J. Klir |
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XV, 381 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a IFSR international series on systems science and engineering |v 25 | |
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| 650 | 4 | |a Measure theory | |
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Contents
1
Introduction
. 1
Notes. 7
2
Preliminaries
. 9
2.1
Classical Sets
. 9
2.1.1
Set Inclusion and Characteristic Function
. 9
2.1.2
Operations on Sets
. 12
2.1.3
Classes of Sets
. 17
2.1.4
Atoms and Holes
. 26
2.1.5
S-Compact Space
. 31
2.1.6
Relations, Posets, and Lattices
. 34
2.2
Classical Measures
. 37
2.3
Fuzzy Sets
. 43
Notes
. 53
Exercises
. 55
3
Basic Ideas of Generalized Measure Theory
. 61
3.1
Generalizing Classical Measures
. 61
3.2
Monotone Measures
. 63
3.3 Superadditive
and
Subadditive
Measures
. 67
3.4
Signed General Measures
. 68
Notes
. 70
Exercises
. 71
4
Special Areas of Generalized Measure Theory
. 73
4.1
An Overview
. 73
4.2
Choquet Capacities
. 74
4.3
Я
-Measures.
77
4.4
Quasi-Measures
. 85
4.5
Belief Measures and Plausibility Measures
. 89
4.6
Possibility Measures and Necessity Measures
. 98
4.7
Properties of Finite Monotone Measures
. 103
Notes
. 104
Exercises
. 106
Extensions
.
Ill
5.1
A General Discussion on Extensions
. 111
5.2
Extension
of Quasi-Measures and
i-Measures
. 112
5.3
Extension
of
Semicontinuous Monotone
Measures
. 116
5.4
Absolute Continuity and Extension of Monotone
Measures
. 120
5.5
Extension of Possibility Measures and Necessity Measures
. 123
Notes
. 130
Exercises
. 130
Structural Characteristics for Set Functions
. 133
6.1
Null-Additivity
. 133
6.2
Autocontinuity
. 135
6.3
Uniform Autocontinuity
. 143
6.4
Structural Characteristics of Monotone Set Functions
. 144
6.5
Monotone Measures on ^-Compact Space
. 147
Notes
. 148
Exercises
. 148
Measurable Functions on Monotone Measure Spaces
. 151
7.1
Measurable Functions
. 151
7.2
"Almost" and "Pseudo-Almost"
. 153
7.3
Relation Among Convergences of Measurable Function
Sequence
. 156
7.4
Convergences of Measurable Function Sequence
on Possibility Measure Spaces
. 162
Notes
. 164
Exercises
. 164
Integration
. 167
8.1
The Lebesgue Integral
. 167
8.2
Properties of the Lebesgue Integral
. 170
8.3
Lebesgue Integrals on Finite Sets
. 172
8.4
A General View of Integration on Finite Sets
. 174
Notes
. 177
Exercises
. 177
Sugeno Integrals
. 179
9.1
Definition
. 179
9.2
Properties of the Sugeno Integral
. 183
9.3
Convergence Theorems of the Sugeno Integral Sequence
. 191
9.4
Transformation Theorem for Sugeno Integrals
. 202
9.5 Monotone
Measures Defined by Sugeno Integrals
. 203
9.6
More Results on Sugeno Integrals with Respect
to a Monotone Measure
. 205
Notes
. 207
Exercises
. 208
10
Pan-Integrals
. 213
10.1
Pan-Additions and Pan-Multiplications
. 213
10.2
Definition of Pan-Integrals
. 214
10.3
Properties of Pan-Integral
. 218
10.4
A Transformation Theorem
. 220
Notes
. 222
Exercises
. 223
11
Choquet Integrals.
. 225
11.1
Choquet Integrals for
Nonnegative
Functions
. 225
11.2
Properties of the Choquet Integral
. 227
11.3
Translatable and Symmetric Choquet Integrals
. 230
11.4
Convergence Theorems
. 234
11.5
Choquet Integrals on Finite Sets
. 239
11.6
An Alternative Calculation Formula
. 243
Notes
. 245
Exercises
. 245
12
Upper and Lower Integrals
. 247
12.1
Definitions
. 247
12.2
Properties
. 249
12.3
Relations Between Integrals
. 256
12.4
Lower and Upper Integrals on Finite Sets
. 265
12.5
Uncertainty Carried by Monotone Measures
. 269
Notes
. 272
Exercises
. 273
13
Constructing General Measures
. 275
13.1
An Overview
. 275
13.2
Constructing New Measures via Integration
. 276
13.3
Constructing New Measures by Transformations
. 277
13.4
Constructing New Measures by Identification
and Extension
. 278
13.5
Data-Driven Construction Methods
. 279
13.6
Other Construction Methods
. 282
13.6.1
Methods Based on Modal Logic
. 282
13.6.2
Methods Based on Uncertainty Principles
. 283
Notes
. 284
14
Fuzzification of Generalized Measures and the Choquet Integral.
. 285
14.1
Conventions
. 285
14.2
Monotone Measures Defined on Fuzzy
σ
-Algebras
.
285
14.3
The Choquet Extension
. 286
14.4
Structural Characteristics of Monotone Measures
on Fuzzy -Algebras
. 287
14.5
Hereditability of Structural Characteristics
. 289
14.6
Real-Valued Choquet Integrals with Fuzzy-Valued
Integrands
. 294
Notes
. 302
15
Applications of Generalized Measure Theory
.
χ
303
15.1
General Remarks
. 303
15.2
Generalized Information Theory: An Overview
. 305
15.3
Theories of Imprecise Probabilities
. 307
15.4
Classification of Pairs of Dual Measures
. 313
15.5
Utility of Some Special Theories of Imprecise
Probabilities
. 318
15.5.1
Dempster-Shafer Theory
(DST)
. 318
15.5.2
Possibility Theory
. 324
15.6
Information Fusion
. 326
15.7
Multiregression
. 333
15.8
Classification
. 335
15.9
Other Applications: An Overview
. 337
Notes
. 338
Exercises
. 340
Appendix A
. 343
Appendix
В
. 351
Bibliography
. 355
Subject Index
. 373
Name Index
. 379 |
| adam_txt |
Contents
1
Introduction
. 1
Notes. 7
2
Preliminaries
. 9
2.1
Classical Sets
. 9
2.1.1
Set Inclusion and Characteristic Function
. 9
2.1.2
Operations on Sets
. 12
2.1.3
Classes of Sets
. 17
2.1.4
Atoms and Holes
. 26
2.1.5
S-Compact Space
. 31
2.1.6
Relations, Posets, and Lattices
. 34
2.2
Classical Measures
. 37
2.3
Fuzzy Sets
. 43
Notes
. 53
Exercises
. 55
3
Basic Ideas of Generalized Measure Theory
. 61
3.1
Generalizing Classical Measures
. 61
3.2
Monotone Measures
. 63
3.3 Superadditive
and
Subadditive
Measures
. 67
3.4
Signed General Measures
. 68
Notes
. 70
Exercises
. 71
4
Special Areas of Generalized Measure Theory
. 73
4.1
An Overview
. 73
4.2
Choquet Capacities
. 74
4.3
Я
-Measures.
77
4.4
Quasi-Measures
. 85
4.5
Belief Measures and Plausibility Measures
. 89
4.6
Possibility Measures and Necessity Measures
. 98
4.7
Properties of Finite Monotone Measures
. 103
Notes
. 104
Exercises
. 106
Extensions
.
Ill
5.1
A General Discussion on Extensions
. 111
5.2
Extension
of Quasi-Measures and
i-Measures
. 112
5.3
Extension
of
Semicontinuous Monotone
Measures
. 116
5.4
Absolute Continuity and Extension of Monotone
Measures
. 120
5.5
Extension of Possibility Measures and Necessity Measures
. 123
Notes
. 130
Exercises
. 130
Structural Characteristics for Set Functions
. 133
6.1
Null-Additivity
. 133
6.2
Autocontinuity
. 135
6.3
Uniform Autocontinuity
. 143
6.4
Structural Characteristics of Monotone Set Functions
. 144
6.5
Monotone Measures on ^-Compact Space
. 147
Notes
. 148
Exercises
. 148
Measurable Functions on Monotone Measure Spaces
. 151
7.1
Measurable Functions
. 151
7.2
"Almost" and "Pseudo-Almost"
. 153
7.3
Relation Among Convergences of Measurable Function
Sequence
. 156
7.4
Convergences of Measurable Function Sequence
on Possibility Measure Spaces
. 162
Notes
. 164
Exercises
. 164
Integration
. 167
8.1
The Lebesgue Integral
. 167
8.2
Properties of the Lebesgue Integral
. 170
8.3
Lebesgue Integrals on Finite Sets
. 172
8.4
A General View of Integration on Finite Sets
. 174
Notes
. 177
Exercises
. 177
Sugeno Integrals
. 179
9.1
Definition
. 179
9.2
Properties of the Sugeno Integral
. 183
9.3
Convergence Theorems of the Sugeno Integral Sequence
. 191
9.4
Transformation Theorem for Sugeno Integrals
. 202
9.5 Monotone
Measures Defined by Sugeno Integrals
. 203
9.6
More Results on Sugeno Integrals with Respect
to a Monotone Measure
. 205
Notes
. 207
Exercises
. 208
10
Pan-Integrals
. 213
10.1
Pan-Additions and Pan-Multiplications
. 213
10.2
Definition of Pan-Integrals
. 214
10.3
Properties of Pan-Integral
. 218
10.4
A Transformation Theorem
. 220
Notes
. 222
Exercises
. 223
11
Choquet Integrals.
. 225
11.1
Choquet Integrals for
Nonnegative
Functions
. 225
11.2
Properties of the Choquet Integral
. 227
11.3
Translatable and Symmetric Choquet Integrals
. 230
11.4
Convergence Theorems
. 234
11.5
Choquet Integrals on Finite Sets
. 239
11.6
An Alternative Calculation Formula
. 243
Notes
. 245
Exercises
. 245
12
Upper and Lower Integrals
. 247
12.1
Definitions
. 247
12.2
Properties
. 249
12.3
Relations Between Integrals
. 256
12.4
Lower and Upper Integrals on Finite Sets
. 265
12.5
Uncertainty Carried by Monotone Measures
. 269
Notes
. 272
Exercises
. 273
13
Constructing General Measures
. 275
13.1
An Overview
. 275
13.2
Constructing New Measures via Integration
. 276
13.3
Constructing New Measures by Transformations
. 277
13.4
Constructing New Measures by Identification
and Extension
. 278
13.5
Data-Driven Construction Methods
. 279
13.6
Other Construction Methods
. 282
13.6.1
Methods Based on Modal Logic
. 282
13.6.2
Methods Based on Uncertainty Principles
. 283
Notes
. 284
14
Fuzzification of Generalized Measures and the Choquet Integral.
. 285
14.1
Conventions
. 285
14.2
Monotone Measures Defined on Fuzzy
σ
-Algebras
.
285
14.3
The Choquet Extension
. 286
14.4
Structural Characteristics of Monotone Measures
on Fuzzy -Algebras
. 287
14.5
Hereditability of Structural Characteristics
. 289
14.6
Real-Valued Choquet Integrals with Fuzzy-Valued
Integrands
. 294
Notes
. 302
15
Applications of Generalized Measure Theory
.
χ
303
15.1
General Remarks
. 303
15.2
Generalized Information Theory: An Overview
. 305
15.3
Theories of Imprecise Probabilities
. 307
15.4
Classification of Pairs of Dual Measures
. 313
15.5
Utility of Some Special Theories of Imprecise
Probabilities
. 318
15.5.1
Dempster-Shafer Theory
(DST)
. 318
15.5.2
Possibility Theory
. 324
15.6
Information Fusion
. 326
15.7
Multiregression
. 333
15.8
Classification
. 335
15.9
Other Applications: An Overview
. 337
Notes
. 338
Exercises
. 340
Appendix A
. 343
Appendix
В
. 351
Bibliography
. 355
Subject Index
. 373
Name Index
. 379 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV023366249 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:10:55Z |
| indexdate | 2024-07-20T09:43:15Z |
| institution | BVB |
| isbn | 0387768513 9780387768519 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016549602 |
| oclc_num | 192027292 |
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| physical | XV, 381 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | IFSR international series on systems science and engineering |
| series2 | IFSR international series on systems science and engineering |
| spelling | Wang, Zhenyuan Verfasser (DE-588)13782503X aut Generalized measure theory Zhenyuan Wang ; George J. Klir New York, NY Springer 2009 XV, 381 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier IFSR international series on systems science and engineering 25 Integrals, Generalized Measure theory Maßtheorie (DE-588)4074626-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 s DE-604 Klir, George J. Verfasser aut Erscheint auch als Online-Ausgabe 978-0-387-76852-6 IFSR international series on systems science and engineering 25 (DE-604)BV000016922 25 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3015565&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016549602&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wang, Zhenyuan Klir, George J. Generalized measure theory IFSR international series on systems science and engineering Integrals, Generalized Measure theory Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4074626-4 |
| title | Generalized measure theory |
| title_auth | Generalized measure theory |
| title_exact_search | Generalized measure theory |
| title_exact_search_txtP | Generalized measure theory |
| title_full | Generalized measure theory Zhenyuan Wang ; George J. Klir |
| title_fullStr | Generalized measure theory Zhenyuan Wang ; George J. Klir |
| title_full_unstemmed | Generalized measure theory Zhenyuan Wang ; George J. Klir |
| title_short | Generalized measure theory |
| title_sort | generalized measure theory |
| topic | Integrals, Generalized Measure theory Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Integrals, Generalized Measure theory Maßtheorie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3015565&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016549602&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000016922 |
| work_keys_str_mv | AT wangzhenyuan generalizedmeasuretheory AT klirgeorgej generalizedmeasuretheory |