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:

1. When E is the empty set then $$g(E)=0$$.

2. When E is a subset of F, then $$g(E)\leq g(F)$$.

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

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

Definition
Let $$X = \left\lbrace x_1,...,x_n \right\rbrace $$  be a finite set and let $$\lambda \in (-1,+\infty)$$. A Sugeno $$\lambda$$-measure is a function g from $$2^X$$ to [0, 1] with properties:


 * 1) $$g(X) = 1$$.
 * 2) if A, B $$\subseteq 2^X$$ 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 $$\left\lbrace x_i \right\rbrace $$ is called a density and is denoted by $$g_i = g(\left\lbrace x_i \right\rbrace)$$. In addition, we have that $$\lambda$$ satisﬁes the property

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

Tahani and Keller 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 $$\lambda$$ uniquely.