Fuzzy measures assuming their values in the set of fuzzy numbers (Q1063131)

A fuzzy set on a non-empty set X is introduced as a mapping of X into [0,1] and a fuzzy \(\sigma\)-algebra \(\sigma\) on \([0,1]^ X\) is defined to be a system of subsets of \([0,1]^ X\) closed under complementation and countable union, containing all constant fuzzy sets. The family \(\zeta\) (\({\mathfrak A})\) of all \({\mathfrak A}\)-measurable, [0,1]-valued functions forms a fuzzy \(\sigma\)-algebra, where \({\mathfrak A}\) is a classical \(\sigma\)- algebra. A finite fuzzy number is a distribution function F on the extended non-negative real numbers, where \(F^{-1}\) denotes the corresponding inverse defined by \(F^{-1}(a)=\sup \{r\geq 0:F(r)<a\}\). Finally a finite fuzzy-valued fuzzy measure m is a mapping of a fuzzy \(\sigma\)-algebra \(\sigma\) into the set of all distribution functions on the extended non-negative real numbers such that \(m(1)<\delta_{\infty};(m(0))^{-1}=0;(m(\mu \bigvee \nu))^{- 1}+(m(\mu \bigwedge \nu))^{-1}=(m(\mu))^{-1}+(m(\nu))^{-1};\mu_ n\uparrow\) implies (m(\(\bigvee_{n\in {\mathbb{N}}}\mu_ n))^{- 1}=\bigvee_{n\in {\mathbb{N}}}(m(\mu_ n))^{-1}.\) The main result of the article concerns a characterization of finite fuzzy valued fuzzy measures defined on a fuzzy \(\sigma\)-algebra \(\zeta\) (\({\mathfrak A})\) on \([0,1]^ X\) with values in the sets of all distribution functions on the non-negative real numbers by means of an integral representation (relative to a so-called adapted family, consisting of a system \((P_{\alpha},K_{\alpha})_{\alpha \in [0,1]}\) of classical probability measures \(P_{\alpha}\) and Markov kernels \(K_{\alpha})\), which looks as follows: \[ (m(\mu))^{- 1}(\alpha)=\int_{X}K_{\alpha}(x,[0,\mu (x)])dP_{\alpha}(x). \]