Random sets and Choquet-type representations

Abstract

As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decomposable combinations and Choquet convex decomposable combinations, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull, without any topological or algebraic assumptions on the underlying set. Furthermore, we show that the Choquet decomposable hull of a set coincides with its strongly closed decomposable hull and the Choquet convex decomposable hull of a set coincides with the Choquet decomposable hull of its convex hull. It turns out that the measurable selections of a closed-valued multifunction form a Choquet decomposable set and those of a closed convex-valued multifunction form a Choquet convex decomposable set. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.