27-09-2014, 11:04 AM
fuzzy measure theory
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INTRODUCTION
The principal purpose of this project is to present a relatively new mathematical subject referred to as fuzzy measure theory . The emergence of fuzzy measure theory (in the late 1970’s) exemplifies a significant trend toward generalizations of existing mathematical concepts and theories . Each generalization enriches not only our insights but also our capabilities to properly model the problems of real world.
Upon consulting a common dictionary, we can find several meanings for uncertain .if we examine these meanings , two categories of uncertainty emerge : vagueness and ambiguity , respectively. In general , vagueness is associated with the difficulty of making sharp or precise distinctions in the world, on the other hand, ambiguity is associated with the situations in which the choice between two or more alternatives is unspecified.
Each of these two distinct forms of uncertainty -vagueness and ambiguity - is connected with a set of related concepts . some of the concepts connected with vagueness are fuzziness, haziness , cloudiness, unclearness, indistinctiveness and sharplessness; some of the concepts connected with ambiguity are nonspecificity, one-to m-many relation, variety, generality, diversity and divergence. A mathematical formulation of these various types of uncertainty is now available in terms of the theory of fuzzy sets and fuzzy measures . The concept of a fuzzy set provides a basic mathematical framework for dealing with vagueness. On the other
hand , the concept of a fuzzy measure provides a general framework for dealing with ambiguity.
Fuzzy measure theory is an offspring of classical measure theory. The latter has its roots in metric geometry , which is characterized by assigning numbers to lengths , areas or volumes. This measurement was first conceived simply as a comparison with a standard unit. Soon it is revealed that measurement is more complicated than this simple process. Integral calculus , based upon the Riemann integral , was the first tool to deal with the problem.
Classical measure theory is based on a measure space (X,A,m), where A is a σ-algebra of subsets of a nonempty set X and m is a nonnegative σ -additive real set function defined on A. Recall that (X ,A ) is a measurable space and m is called a measure .if m(X)=1, then m is called a probability measure. hence , probability theory may be viewed as a part of classical measure theory. The concept of a probability measure was formulated axiomatically in 1933 by Andrei N.Kolmogrov, a Russian mathematician
There are four chapters in this project, describing the different types of fuzzy measures. In first chapter, we give some definitions and results useful in the following chapters.
In chapter II , we give the definitions and examples of fuzzy measures and semicontinuous fuzzy measure.
In chapter III ,we attempt to construct two types of nonadditive measures, namely, belief measure and plausibility measure in the context of probability theory.
CONCLUSION
This paper adopts the concept of fuzzy measure. we know about basic foundations of the theory of fuzzy measure as they can be considered undisputed as of today. The principal theme of this project , fuzzy measure , also emerged in the context of fuzzy sets . The concept of can also be generalised by new concepts of measure that pretend to measure a characteristic not really related with the inclusion of sets. The concept of fuzzy measure does not require additivity, but it requires monotonicity related to the inclusion of sets.