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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. Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... The membership function of a fuzzy set corresponds to the indicator function of classical sets. ... Probability is the likelihood that something is the case or will happen. ...

1 Axioms
2 Examples of Fuzzy Measures
- 2.1 Sugeno λ-measure
  - 2.1.1 Definition
3 See also
4 External links
5 References

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: In mathematics, the definition of the probability space is the foundation of probability theory. ...

1. When E is the empty set then $g (E) = 0$ . The empty set is the set containing no elements. ...

2. When E is a subset of F, then $g(E)leq g(F)$ . â€œSupersetâ€ redirects here. ...

A fuzzy measure g is called normalized if $g (X) = 1$ .

Examples of Fuzzy Measures

Sugeno $λ$ -measure

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

Definition

Let $X = leftlbrace x_1,...,x_n rightrbrace$ be a finite set and let $lambda in (-1,+infty)$ . A Sugeno $λ$ -measure is a function g from $2 X$ to [0, 1] with properties:

$g (X) = 1$ .
if A, B with $A cap B = emptyset$ then $g(A cup B) =g(A)+g(B)+lambda g(A)g(B)$ .

As a convention, the value of g at a singleton set $leftlbrace x_i rightrbrace$ is called a density and is denoted by $g_i = g(leftlbrace x_i rightrbrace)$ . In addition, we have that $λ$ satisﬁes the property

$lambda +1 = prod_{i=1}^n (1+lambda g_i)$ .

Tahani and Keller ^[1]as well as Wang and Klir have showed that that once the densities are known, it is possible to use the previous polynomial to obtain the values of $λ$ uniquely. In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

External links

http://pami.uwaterloo.ca/tizhoosh/measure.htm

References

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

^ 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.

Categories: Mathematics articles needing expert attention | Articles needing expert attention | Statistics stubs | Probability theory | Fuzzy logic 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 Binghamton University in New York. ...

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