Random sets and Choquet-type representations

TL;DR

This paper introduces and analyzes Choquet combinations and hulls as generalizations of convex combinations, exploring their properties in finite and infinite-dimensional Lebesgue-Bochner spaces and their relation to decomposable hulls.

Contribution

It defines Choquet hulls and decompositions, compares them with convex hulls, and studies their properties and operator features in Lebesgue-Bochner spaces, extending convex analysis.

Findings

Choquet hull equals convex hull in finite dimensions

Choquet hull can be larger than convex hull in infinite dimensions

Measurable selections are Choquet decomposable

Abstract

As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decompositions and Choquet convex decompositions, 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. Furthermore, we show that Choquet decomposable hull of a set coincides with its (strongly) closed decomposable hull and the Choquet convex decomposable hull of a set coincides with its Choquet decomposable hull of the convex hull. It turns out that the collection of all measurable selections of a closed-valued multifunction is Choquet decomposable and those of a closed convex-valued multifunction is Choquet convex decomposable. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.