Choquet integrals of set-valued functions with respect to set-valued fuzzy measures

Abstract

As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set-valued fuzzy measures. First, a set-valued Choquet integral of a real-valued function is defined using the set of Choquet integrals of real-valued functions with respect to fuzzy measure selections of a set-valued fuzzy measure. The integrand is then extended to set-valued functions, after which various properties, convergence theorems, and set-valued Jensen's inequalities are obtained. Second, following how single-valued Choquet integrals are defined, an alternative set-valued Choquet integral of real-valued function is formulated using Aumann integrals. Through these steps, the existing set-valued Choquet integrals are covered.