On the equivalence of Aumann and Herer expectations of random sets

Abstract

Spaces where the Aumann and Herer notions of expectation of a random set coincide are exactly those having the Mazur Intersection Property (the closed convex hull of a bounded set is the intersection of all balls covering it). For a random compact set, more can be said: its Herer expectation is always the intersection of all closed balls covering its Aumann expectation. Some further consequences of these results are presented.