# Fuzzy measure theory

**Fuzzy measure theory**considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used. The central concept of fuzzy measure theory is fuzzy measure, which was introduced by Sugeno in 1974.

## Axioms

Fuzzy measure can be considered as generalization of the classical probability measure. A fuzzy measure*g*over a set

*X*(the universe of discourse with the subsets

*E*,

*F*, ...) satisfies the following conditions when

*X*is finite:

1. When

*E*is the empty set then .

2. When

*E*is a subset of

*F*, then .

A fuzzy measure

*g*is called

*normalized*if .

## Examples of Fuzzy Measures

### Sugeno -measure

The Sugeno -measure is a special case of fuzzy measures defined iteratively. It has the following definition#### Definition

Let be a finite set and let . A**Sugeno -measure**is a function

*g*from to [0, 1] with properties:

- .
- if
*A*,*B*with then .

As a convention, the measure of a singleton set is called a density and is denoted by . In addition, we have that satisﬁes the property

.

Tahani and Keller

^{[1]}have showed that that once the densities are known, it is possible to use the previous polynomial to obtain the values of .

## See also

## External links

## References

- Wang, Zhenyuan, and , George J. Klir,
*Fuzzy Measure Theory*, Plenum Press, New York, 1991.

1. ^ H. Tahani and J. Keller (1990). "Information Fusion in Computer Vision Using the Fuzzy Integral".

*IEEE Transactions on Systems, Man and Cybernetic***20**: 733-741.**Fuzzy sets**are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent

**.....**Click the link for more information.

The

**membership function**of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation.**.....**Click the link for more information.**Probability**is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of

**.....**Click the link for more information.

**empty set**is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set. The empty set is also sometimes called the

**null set**

**.....**Click the link for more information.

**subset**of a set

*B*if

*A*is "contained" inside

*B*. Notice that

*A*and

*B*may coincide. The relationship of one set being a subset of another is called

**inclusion**or

**containment**.

**.....**Click the link for more information.

In mathematics, a

**polynomial**is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. is a polynomial.**.....**Click the link for more information.**Probability theory**is the branch of mathematics concerned with analysis of random phenomena.

^{[1]}The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities

**.....**Click the link for more information.

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**Possibility theory**is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets**.....**Click the link for more information.**George Jiri Klir**(born 1932 Prague, Czechoslovakia) is a Czech-American computer scientist and professor of systems sciences at the Center for Intelligent Systems at the State University of New York at Binghamton, New York.

**.....**Click the link for more information.

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