Related to many applications in different fields, such as game theory, information fusion, data mining, and decision making, we have introduced in one our previous paper so called generalized Choquet-type integral for a real-valued function concerning a set-function and a σ-additive measure. The present study further generalizes the generalized Choquet-type integral in terms of a double set-function Choquet integral for a real-valued function based on a set-function and fuzzy measure. Several of its properties and convergence theorems are obtained, and a novel type of Jensen's inequality is proved. The stability of the proposed system formed by a double set-function Choquet integral concerning multiple inputs and one output is indicated. An effective application in decision making is shown through numerical examples.