The Hausdorff Metric Topology, the Attouch-Wets Topology, and the Measurability of Set-Valued Functions

Let ⟨X, d⟩ be a separable metric space. A set-valued function Γ: S ⇉ X defined on a measurable space S whose values are nonempty closed subsets of X is declared measurable provided for each open subset V of X, {s ∈ S: Γ(s) ∩ V ≠ ∅} is a measurable subset of S. In this paper, we look at the relationship between measurability so defined and the Borel measurability of Γ, viewed as a single-valued function into the nonempty closed subsets of X, equipped with either the Hausdorff metric topology or with the Attouch-Wets topology. Our analysis rests on cardinality arguments in conjunction with the representation of these hyperspaces as weak topologies. Applications are given to convex-valued multifunctions.