On fuzzy measures defined by fuzzy integrals (Q1105726)

A fuzzy measure is defined to be a set function \(\mu\) defined on a \(\sigma\)-algebra \({\mathcal F}\) of subsets of X which is monotone and continuous from above and below. The family of functions FM(\(\mu)\) denotes the set of those non-negative measurable functions f which satisfy \(\mu \{x\in X:\alpha \leq f(x)\leq \beta \}<\infty\) for any \(\alpha\) and \(\beta\) with \(0<\alpha <\beta;\) for an \({\mathcal F}\)-measurable non-negative extended real-valued function on X, \(\lambda_ f(E)\) is defined to be equal to \(\sup_{0\leq \alpha <\infty}[\alpha \wedge \mu (\{x\in X:f(x)\geq \alpha \}\cap E)].\) The important result reached in the paper is as follows: Let \((X,{\mathcal F},\mu)\) be a \(\sigma\)-finite measure space satisfying the condition \(\mu (E)<\infty\) and \(\mu (F)<\infty\) \((E,F\in {\mathcal F})\Rightarrow \mu(E\cup F)<\infty.\) Then the following three statements are equivalent: (1) \(f\in FM(\mu)\), (2) if there exists \(E\in {\mathcal F}\) such that \(\mu (\{x\in X:f(x)\geq \alpha \}\cap E)<\infty\) for some \(\alpha >0,\) then the relation is true for any \(\alpha >0\), (3) the set function \(\lambda_ f\) induced by f is a fuzzy measure. The author studies set function \(\lambda_ f\) with regard to inheritance of properties from the fuzzy measure \(\mu\). Finally, some examples of fuzzy measures obtained by ordinary functions are given.