Convergence of closed convex sets and σ-fields

Abstract

Let {Cn} 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 Cnexists 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. © 1983 Springer-Verlag.