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Generalized Measure Theory

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Springer, Oct 27, 2008 - Mathematics - 381 pages
Generalized Measure Theory examines the relatively new mathematical area of generalized measure theory. The exposition unfolds systematically, beginning 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.
  

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Contents

Chapter 1 Introduction
1
Chapter 2 Preliminaries
9
Chapter 3 Basic Ideas of Generalized Measure Theory
61
Chapter 4 Special Areas of Generalized Measure Theory
73
Chapter 5 Extensions
111
Chapter 6 Structural Characteristics for Set Functions
133
Chapter 7 Measurable Functions on Monotone Measure Spaces
151
Chapter 8 Integration
167
Chapter 12 Upper and Lower Integrals
247
Chapter 13 Constructing General Measures
275
Chapter 14 Fuzzification of Generalized Measures and the Choquet Integral
285
Chapter 15 Applications of Generalized Measure Theory
303
Appendix A
343
Appendix B
351
Bibliography
355
Subject Index
373

Chapter 9 Sugeno Integrals
179
Chapter 10 PanIntegrals
213
Chapter 11 Choquet Integrals
224

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About the author (2008)

GEORGE J. KLIR, PhD, is currently Distinguished Professor of Systems Science at Binghamton University, SUNY. Since immigrating to the U.S. in 1966, he has held positions at UCLA, Fairleigh Dickinson University, and Binghamton University. He is a Life Fellow of IEEE, IFSA, and the Netherlands Institute for Advanced Studies. He has served as president of SGSR, IFSR, NAFIPS, and IFSA. He has published over 300 research papers and sixteen books, and has edited ten books. He has also served as Editor in Chief of the International Journal of General Systems since 1974 and of the IFSR International Book Series on Systems Science and Engineering since 1985. He has received numerous professional awards, including five honorary doctoral degrees, Bernard Bolzano's Gold Medal, Arnold Kaufmann's Gold Medal, and the SUNY Chancellor's Award for "Exemplary Contributions to Research and Scholarship." He is listed in Who's Who in America and Who's Who in the World. His current research interests include intelligent systems, soft computing, generalized information theory, systems modeling and design, fuzzy systems, and the theory of generalized measures. He has guided twenty-nine successful doctoral dissertations in these areas. Some of his research has been funded by grants from NSF, ONR, the United States Air Force, NASA, Sandia Labs, NATO, and various industries.