Joint measurability of mappings induced by a fuzzy random variable

Abstract

The operation of taking the alpha-cut of a compact convex fuzzy set is shown to be jointly measurable with respect to both alpha and the fuzzy set. As a consequence, a number of mappings on product spaces which are induced by a fuzzy random variable are shown to be jointly measurable. Some applications to the relationships between fuzzy random variables and other imprecise random elements are obtained. Finally, a number of conditions are shown to be equivalent to being a fuzzy random variable, at least in the case that the sigma-algebra in the sample space is complete, and logical implications and equivalences between them are established in the general case.