Fuzzy Measure Theory / Edition 1

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Providing the first comprehensive treatment of the subject, this groundbreaking work is solidly founded on a decade of concentrated research, some of which is published here for the first time, as well as practical, ''hands on'' classroom experience. The clarity of presentation and abundance of examples and exercises make it suitable as a graduate level text in mathematics, decision making, artificial intelligence, and engineering courses.

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Product Details

  • ISBN-13: 9780306442605
  • Publisher: Springer US
  • Publication date: 1/31/1993
  • Edition description: 1992
  • Edition number: 1
  • Pages: 354
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.88 (d)

Table of Contents

Ch. 1 Introduction 1
Ch. 2 Required Background in Set Theory 7
2.1 Set Inclusion and Characteristic Function 7
2.2 Operations on Sets 9
2.3 Classes of Sets 15
2.4 Atoms and Holes 22
2.5 S-Compact Spaces 28
2.6 Relations, Posets, and Lattices 30
Ch. 3 Fuzzy Measures 39
3.1 Fuzzy Measures and Semicontinuous Fuzzy Measures 39
3.2 [lambda]-Fuzzy Measures 42
3.3 Quasi-measures 50
3.4 Belief Measures and Plausibility Measures 54
3.5 Possibility Measures and Necessity Measures 62
3.6 Properties of Finite Fuzzy Measures 66
Ch. 4 Extensions 73
4.1 Extension of Quasi-Measures and [lambda]-Fuzzy Measures 73
4.2 Extension of Semicontinuous Fuzzy Measures 78
4.3 Absolute Continuity and Extension of Fuzzy Measures 81
4.4 Extension of Possibility Measures and Necessity Measures 84
Ch. 5 Structural Characteristics for Set Functions 95
5.1 Null-Additivity 95
5.2 Autocontinuity 97
5.3 Uniform Autocontinuity 106
5.4 Structural Characteristics of Monotone Set Functions 107
5.5 Fuzzy Measures on S-Compact Spaces 110
Ch. 6 Measurable Functions on Fuzzy Measure Spaces 115
6.1 Measurable Functions 115
6.2 "Almost" and "Pseudo-Almost" 117
6.3 Relation Among Convergence of Measurable Function Sequence 120
6.4 Convergence of Measurable Function Sequence on Possibility Measure Spaces 126
Ch. 7 Fuzzy Integrals 131
7.1 Definition 131
7.2 Properties of the Fuzzy Integral 135
7.3 Convergence Theorems of the Fuzzy Integral Sequence 143
7.4 Transformation Theorem for Fuzzy Integrals 152
7.5 Fuzzy Measures Defined by Fuzzy Integrals 154
7.6 Fuzzy Integrals with Respect to a Nonnegative Monotone Set Function 157
Ch. 8 Pan-Integrals 163
8.1 Pan-Additions and Pan-Multiplications 163
8.2 Definition of Pan-Integral 164
8.3 Properties of Pan-Integral 168
8.4 A Transformation Theorem 170
Ch. 9 Applications 175
9.1 General Remarks 175
9.2 Dempster-Shafer Theory 176
9.3 Possibility Theory 181
9.4 Fuzzy Measures and Fuzzy Integrals 184
9.5 Fuzzy Convolution of Fuzzy Distributions 190
References 195
Appendix A. Some Concepts and Results Regarding Classical Measures 203
Appendix B. Some Concepts and Results Regarding Fuzzy Sets 209
Appendix C. Glossary of Key Concepts 219
Appendix D. Glossary of Symbols 227
Appendix E. New Directions in Fuzzy Measure Theory 231
Appendix F. Representative Applications of Fuzzy Measure Theory 285
Author Index 343
Subject Index 347
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