Convergence of closed convex sets and ?-fields

Abstract

Let {C n } be a sequence of closed convex subsets in a Hilbert space H. We prove that the prediction sequence {p(x¦Cn)} converges for every xεH if and only if s-lim C n exists and is not empty. We further show the relation between the limit of closed convex sets and the one of σ-subfields in probability measure spaces.