Generalized Measure Theory

ISBN-13: 9780387768519 / Angielski / Twarda / 2008 / 384 str.

Zhenyuan Wang; George J. Klir

Opis

In 1992 we published a book entitled Fuzzy Measure Theory (Plenum Press, New York), in which the term ''fuzzy measure'' was used for set functions obtained by replacing the additivity requirement of classical measures with weaker requirements of monotonicity with respect to set inclusion and con- nuity. That is, the book dealt with nonnegative set functions that were mo- tone, vanished at the empty set, and possessed appropriate continuity properties when defined on infinite sets. It seems that Fuzzy Measure Theory was the only book available on the market at that time devoted to this emerging new mathematical theory. Some ten years after its publication we began to see that the subject had expanded so much that a second edition of the book, or even a new book on the subject, was needed. We eventually decided to write a new book because the new material we wished to include was too extensive for--and far beyond the usual scope--of a second edition. More importantly, we felt that some fundamental changes regarding this topic's scope and terminology would be desirable and timely.

Recenzje

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)

Spis treści

Preliminaries.- Basic Ideas of Generalized Measure Theory.- Special Areas of Generalized Measure Theory.- Extensions.- Structural Characteristics for Set Functions.- Measurable Functions on Monotone Measure Spaces.- Integration.- Sugeno Integrals.- Pan-Integrals.- Choquet Integrals.- Upper and Lower Integrals.- Constructing General Measures.- Fuzzification of Generalized Measures and the Choquet Integral.- Applications of Generalized Measure Theory.

O autorze

George J. Klir is currently Distinguished Professor of Systems Science at Binghamton University, SUNY. Since he 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.

Blurb

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.

About the Authors:

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. He has published one book and many papers in these areas.

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.