Radon-Nikodým theorems for generalized fuzzy number measures

Abstract

In this paper we deal with generalized fuzzy number measures defined on Σ and taking values in the set $F_{cwk}\left(X\right)$, where $(\Omega ,\Sigma ,\mu )$ is a complete finite measure space, $cwk\left(X\right)$ is the family of all non-empty convex weakly compact subsets of a Banach space X and $F_{cwk}\left(X\right)$ is such that a fuzzy set $u:X\to [0,1]$ is a member of $F_{cwk}\left(X\right)$ if and only if${\left[u\right]}^{\alpha }\in cwk\left(X\right)\text{ for all }\alpha \in (0,1]({\left[u\right]}^{\alpha }=\{x\in X:u\left(x\right)\ge \alpha \left\}\right).$ We show two Radon-Nikodým theorems for such generalized fuzzy number measures in terms of Pettis integral of fuzzy mappings. The first theorem works for a μ-continuous generalized fuzzy number measure $M:\Sigma \to F_{cwk}\left(X\right)$ of bounded variation in a Banach space X with the Radon-Nikodým property (RNP). The next theorem shows that if a generalized fuzzy number measure $M:\Sigma \to F_{cwk}\left(X\right)$ is “dominated” by a generalized fuzzy number $Q\in F_{cwk}\left(X\right)$ in terms of α-level sets, i.e.,${\left[M\right(A\left)\right]}^{\alpha }\subset \mu \left(A\right){\left[Q\right]}^{\alpha }\text{ for all }A\in \Sigma ,\alpha \in (0,1],$ then there exists a fuzzy mapping $\Gamma :\Omega \to F_{cwk}\left(X\right)$ Pettis integrable with $M\left(A\right)=\left(FP\right){\int }_A\Gamma \left(t\right)d\mu$ for all $A\in \Sigma$, where $\left(FP\right){\int }_A\Gamma \left(t\right)d\mu$ is Pettis integral of fuzzy mapping Γ over A. The main advantage of our results is the absence of any separability assumptions.