Generalized Measure Theory

This 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 associated integration theory.

This textbook provides an exhaustive review of generalized measures, from the theoretical background to the latest applications. It is designed as a main text for graduate courses on Generalized Measure Theory, Fuzzy Measure Theory, Theory of Nonadditive Measures and Theory of Monotone Measures. It can also be used as a reference in courses on Fuzzy Sets, Fuzzy Control and Fuzzy Logic.

Most up-to-date textbook in the field, covering recent applications Numerous examples motivate the theory and exercises at the end of the chapter provide practice to the student Useful to an interdisciplinary cadre of mathematicians, engineers, computer scientists and specialists working in decision making Includes supplementary material: sn.pub/extras

Texte du rabat

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. Kliris 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.

Contenu
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.