A direct proof of the characterization of the convexity of the discrete Choquet integral

Abstract This article presents the first self-contained and direct proof of a widely recognized result that the discrete Choquet integral, when defined from a discrete fuzzy measure (or capacity), is convex if and only if the discrete fuzzy measure itself is submodular. In contrast to existing proofs, our argument is constructed directly from the fuzzy measure defined on a finite set, employing only standard techniques from the theory of capacities and Choquet integration.