Generalized Measure Theory

by Wang, Zhenyuan; Klir, George J.

• ISBN13: 9780387768519
• ISBN10: 0387768513
• Format: Hardcover
• Copyright: 2008-10-10
• Publisher: Springer Verlag

Summary

This textbook is a significantly revised follow-up to the author's 1993 text, Fuzzy Measure Theory, and is its replacement in graduate courses on Generalized Measure Theory, Fuzzy Measure Theory, Theory of Nonadditive and Monotone Measures. It is an excellent overview of the theory and state-of-the-art applications of generalized/fuzzy measures. It ranges from two introductory chapters covering the necessary background to the fuzzy, choquet and pan-integrals as well as the construction and applications of general measure theory. The text ends with useful glossaries of key concepts and symbols.

Author Biography

Zhenyuan Wang is currently a Professor in the Department of Mathematics of University of Nebraska at Omaha. His research interests have been in the areas of nonadditive measures, nonlinear integrals, probability and statistics, and data mining George J. Klir is currently a Distinguished Professor of Systems Science at Binghamton University (SUNY at Binghamton). He has published 29 books and well over 300 papers in a wide range of areas. His current research interests are primarily in the areas of fuzzy systems, soft computing, and generalized information theory

Table of Contents

Introduction	p. 1
Notes	p. 7
Preliminaries	p. 9
Classical Sets	p. 9
Set Inclusion and Characteristic Function	p. 9
Operations on Sets	p. 12
Classes of Sets	p. 17
Atoms and Holes	p. 26
S-Compact Space	p. 31
Relations, Posets, and Lattices	p. 34
Classical Measures	p. 37
Fuzzy Sets	p. 43
Notes	p. 53
Exercises	p. 55
Basic Ideas of Generalized Measure Theory	p. 61
Generalizing Classical Measures	p. 61
Monotone Measures	p. 63
Superadditive and Subadditive Measures	p. 67
Signed General Measures	p. 68
Notes	p. 70
Exercises	p. 71
Special Areas of Generalized Measure Theory	p. 73
An Overview	p. 73
Choquet Capacities	p. 74
[lambda]-Measures	p. 77
Quasi-Measures	p. 85
Belief Measures and Plausibility Measures	p. 89
Possibility Measures and Necessity Measures	p. 98
Properties of Finite Monotone Measures	p. 103
Notes	p. 104
Exercises	p. 106
Extensions	p. 111
A General Discussion on Extensions	p. 111
Extension of Quasi-Measures and [lambda]-Measures	p. 112
Extension of Semicontinuous Monotone Measures	p. 116
Absolute Continuity and Extension of Monotone Measures	p. 120
Extension of Possibility Measures and Necessity Measures	p. 123
Notes	p. 130
Exercises	p. 130
Structural Characteristics for Set Functions	p. 133
Null-Additivity	p. 133
Autocontinuity	p. 135
Uniform Autocontinuity	p. 143
Structural Characteristics of Monotone Set Functions	p. 144
Monotone Measures on S-Compact Space	p. 147
Notes	p. 148
Exercises	p. 148
Measurable Functions on Monotone Measure Spaces	p. 151
Measurable Functions	p. 151
"Almost" and "Pseudo-Almost"	p. 153
Relation Among Convergences of Measurable Function Sequence	p. 156
Convergences of Measurable Function Sequence on Possibility Measure Spaces	p. 162
Notes	p. 164
Exercises	p. 164
Integration	p. 167
The Lebesgue Integral	p. 167
Properties of the Lebesgue Integral	p. 170
Lebesgue Integrals on Finite Sets	p. 172
A General View of Integration on Finite Sets	p. 174
Notes	p. 177
Exercises	p. 177
Sugeno Integrals	p. 179
Definition	p. 179
Properties of the Sugeno Integral	p. 183
Convergence Theorems of the Sugeno Integral Sequence	p. 191
Transformation Theorem for Sugeno Integrals	p. 202
Monotone Measures Defined by Sugeno Integrals	p. 203
More Results on Sugeno Integrals with Respect to a Monotone Measure	p. 205
Notes	p. 207
Exercises	p. 208
Pan-Integrals	p. 213
Pan-Additions and Pan-Multiplications	p. 213
Definition of Pan-Integrals	p. 214
Properties of Pan-Integral	p. 218
A Transformation Theorem	p. 220
Notes	p. 222
Exercises	p. 223
Choquet Integrals	p. 225
Choquet Integrals for Nonnegative Functions	p. 225
Properties of the Choquet Integral	p. 227
Translatable and Symmetric Choquet Integrals	p. 230
Convergence Theorems	p. 234
Choquet Integrals on Finite Sets	p. 239
An Alternative Calculation Formula	p. 243
Notes	p. 245
Exercises	p. 245
Upper and Lower Integrals	p. 247
Definitions	p. 247
Properties	p. 249
Relations Between Integrals	p. 256
Lower and Upper Integrals on Finite Sets	p. 265
Uncertainty Carried by Monotone Measures	p. 269
Notes	p. 272
Exercises	p. 273
Constructing General Measures	p. 275
An Overview	p. 275
Constructing New Measures via Integration	p. 276
Constructing New Measures by Transformations	p. 277
Constructing New Measures by Identification and Extension	p. 278
Data-Driven Construction Methods	p. 279
Other Construction Methods	p. 282
Methods Based on Modal Logic	p. 282
Methods Based on Uncertainty Principles	p. 283
Notes	p. 284
Fuzzification of Generalized Measures and the Choquet Integral	p. 285
Conventions	p. 285
Monotone Measures Defined on Fuzzy [sigma]-Algebras	p. 285
The Choquet Extension	p. 286
Structural Characteristics of Monotone Measures on Fuzzy-Algebras	p. 287
Hereditability of Structural Characteristics	p. 289
Real-Valued Choquet Integrals with Fuzzy-Valued Integrands	p. 294
Notes	p. 302
Applications of Generalized Measure Theory	p. 303
General Remarks	p. 303
Generalized Information Theory: An Overview	p. 305
Theories of Imprecise Probabilities	p. 307
Classification of Pairs of Dual Measures	p. 313
Utility of Some Special Theories of Imprecise Probabilities	p. 318
Dempster-Shafer Theory (DST)	p. 318
Possibility Theory	p. 324
Information Fusion	p. 326
Multiregression	p. 333
Classification	p. 335
Other Applications: An Overview	p. 337
Notes	p. 338
Exercises	p. 340
Appendix A	p. 343
Appendix B	p. 351
Bibliography	p. 355
Subject Index	p. 373
Name Index	p. 379
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