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Generalized Measure Theory : IFSR International Series in Systems Science and Systems Engineering - Zhenyuan Wang

Generalized Measure Theory

IFSR International Series in Systems Science and Systems Engineering

Hardcover

Published: 27th October 2008
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This comprehensive text examines the relatively new mathematical area of generalized measure theory. This area expands classical measure theory by abandoning the requirement of additivity and replacing it with various weaker requirements. Each of these weaker requirements characterizes a class of nonadditive measures. This results in new concepts and methods that allow us to deal with many problems in a more realistic way. For example, it allows us to work with imprecise probabilities.

The exposition of generalized measure theory unfolds systematically. It begins with preliminaries and new concepts, followed by a detailed treatment of important new results regarding various types of nonadditive measures and the associated integration theory. The latter involves several types of integrals: Sugeno integrals, Choquet integrals, pan-integrals, and lower and upper integrals. All of the topics are motivated by numerous examples, culminating in a final chapter on applications of generalized measure theory.

Some key features of the book include: many exercises at the end of each chapter along with relevant historical and bibliographical notes, an extensive bibliography, and name and subject indices. The work is suitable for a classroom setting at the graduate level in courses or seminars in applied mathematics, computer science, engineering, and some areas of science. A sound background in mathematical analysis is required. Since the book contains many original results by the authors, it will also appeal to researchers working in the emerging area of generalized measure theory.

From the reviews:

"The monograph contains many original results of the authors and provides an excellent introduction to the emerging field of generalized measure theory. The presentation of the large material is given with didactical skill, numerous examples serve for motivation and illustration of results. ... easy to read by everyone with a solid background in mathematical analysis, and on this basis in particular for students in mathematics and engineering at the graduate level. ... this book can serve as a starting point for research in this area." (Jan Stankiewicz, Zentralblatt MATH, Vol. 1184, 2010)

Introductionp. 1
Notesp. 7
Preliminariesp. 9
Classical Setsp. 9
Set Inclusion and Characteristic Functionp. 9
Operations on Setsp. 12
Classes of Setsp. 17
Atoms and Holesp. 26
S-Compact Spacep. 31
Relations, Posets, and Latticesp. 34
Classical Measuresp. 37
Fuzzy Setsp. 43
Notesp. 53
Exercisesp. 55
Basic Ideas of Generalized Measure Theoryp. 61
Generalizing Classical Measuresp. 61
Monotone Measuresp. 63
Superadditive and Subadditive Measuresp. 67
Signed General Measuresp. 68
Notesp. 70
Exercisesp. 71
Special Areas of Generalized Measure Theoryp. 73
An Overviewp. 73
Choquet Capacitiesp. 74
[lambda]-Measuresp. 77
Quasi-Measuresp. 85
Belief Measures and Plausibility Measuresp. 89
Possibility Measures and Necessity Measuresp. 98
Properties of Finite Monotone Measuresp. 103
Notesp. 104
Exercisesp. 106
Extensionsp. 111
A General Discussion on Extensionsp. 111
Extension of Quasi-Measures and [lambda]-Measuresp. 112
Extension of Semicontinuous Monotone Measuresp. 116
Absolute Continuity and Extension of Monotone Measuresp. 120
Extension of Possibility Measures and Necessity Measuresp. 123
Notesp. 130
Exercisesp. 130
Structural Characteristics for Set Functionsp. 133
Null-Additivityp. 133
Autocontinuityp. 135
Uniform Autocontinuityp. 143
Structural Characteristics of Monotone Set Functionsp. 144
Monotone Measures on S-Compact Spacep. 147
Notesp. 148
Exercisesp. 148
Measurable Functions on Monotone Measure Spacesp. 151
Measurable Functionsp. 151
"Almost" and "Pseudo-Almost"p. 153
Relation Among Convergences of Measurable Function Sequencep. 156
Convergences of Measurable Function Sequence on Possibility Measure Spacesp. 162
Notesp. 164
Exercisesp. 164
Integrationp. 167
The Lebesgue Integralp. 167
Properties of the Lebesgue Integralp. 170
Lebesgue Integrals on Finite Setsp. 172
A General View of Integration on Finite Setsp. 174
Notesp. 177
Exercisesp. 177
Sugeno Integralsp. 179
Definitionp. 179
Properties of the Sugeno Integralp. 183
Convergence Theorems of the Sugeno Integral Sequencep. 191
Transformation Theorem for Sugeno Integralsp. 202
Monotone Measures Defined by Sugeno Integralsp. 203
More Results on Sugeno Integrals with Respect to a Monotone Measurep. 205
Notesp. 207
Exercisesp. 208
Pan-Integralsp. 213
Pan-Additions and Pan-Multiplicationsp. 213
Definition of Pan-Integralsp. 214
Properties of Pan-Integralp. 218
A Transformation Theoremp. 220
Notesp. 222
Exercisesp. 223
Choquet Integralsp. 225
Choquet Integrals for Nonnegative Functionsp. 225
Properties of the Choquet Integralp. 227
Translatable and Symmetric Choquet Integralsp. 230
Convergence Theoremsp. 234
Choquet Integrals on Finite Setsp. 239
An Alternative Calculation Formulap. 243
Notesp. 245
Exercisesp. 245
Upper and Lower Integralsp. 247
Definitionsp. 247
Propertiesp. 249
Relations Between Integralsp. 256
Lower and Upper Integrals on Finite Setsp. 265
Uncertainty Carried by Monotone Measuresp. 269
Notesp. 272
Exercisesp. 273
Constructing General Measuresp. 275
An Overviewp. 275
Constructing New Measures via Integrationp. 276
Constructing New Measures by Transformationsp. 277
Constructing New Measures by Identification and Extensionp. 278
Data-Driven Construction Methodsp. 279
Other Construction Methodsp. 282
Methods Based on Modal Logicp. 282
Methods Based on Uncertainty Principlesp. 283
Notesp. 284
Fuzzification of Generalized Measures and the Choquet Integralp. 285
Conventionsp. 285
Monotone Measures Defined on Fuzzy [sigma]-Algebrasp. 285
The Choquet Extensionp. 286
Structural Characteristics of Monotone Measures on Fuzzy-Algebrasp. 287
Hereditability of Structural Characteristicsp. 289
Real-Valued Choquet Integrals with Fuzzy-Valued Integrandsp. 294
Notesp. 302
Applications of Generalized Measure Theoryp. 303
General Remarksp. 303
Generalized Information Theory: An Overviewp. 305
Theories of Imprecise Probabilitiesp. 307
Classification of Pairs of Dual Measuresp. 313
Utility of Some Special Theories of Imprecise Probabilitiesp. 318
Dempster-Shafer Theory (DST)p. 318
Possibility Theoryp. 324
Information Fusionp. 326
Multiregressionp. 333
Classificationp. 335
Other Applications: An Overviewp. 337
Notesp. 338
Exercisesp. 340
Appendix Ap. 343
Appendix Bp. 351
Bibliographyp. 355
Subject Indexp. 373
Name Indexp. 379
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387768519
ISBN-10: 0387768513
Series: IFSR International Series in Systems Science and Systems Engineering
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 384
Published: 27th October 2008
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.29
Weight (kg): 0.76