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Journal of Convex Analysis 21 (2014), No. 4, 1085--1103
Copyright Heldermann Verlag 2014

Expectations of Random Sets in Banach Spaces

Pedro Terán
Escuela Politécnica de Ingenierí, Dep. de Estadística e I.O. y D.M., Universidad de Oviedo, 33071 Gijón, Spain

The Aumann and Herer expectations represent two different approaches to defining the expectation of a random set (analytical versus geometrical). This paper investigates their relationships in the setting of Banach spaces, which are shown to be related to the geometry of the dual unit ball, to the bornological differentiability properties of the norm, and to the kind of sets the random set takes on as values. We also show that both expectations are identical for almost all (in the Baire sense) norms equivalent to any given norm provided the dual is strongly separable, and that in general the Herer expectation is the ball hull of the Aumann expectation for almost all norms. The Aumann expectation can be regarded as a special case of the Herer expectation after an embedding into a suitably larger space. This suggests that the Herer expectation is a suitable extension of Aumann's to non-linear metric spaces.

Keywords: Aumann expectation, Herer expectation, intersection of balls, Mazur Intersection Property, random set, weak* denting point.

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