Skorohod's Representation Theorem for Sets of Probabilities

Abstract

From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zero-one if there exists an E in F, a measurable, onto phi:Omega -> Pi such that for all p in Pi, p(phi^{-1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space,
Delta_M is the set of Borel probabilities on M and Comp(
Delta_M) is the class of non-empty, compact subsets of Delta_
M with the Hausdor ff metric. There exists a jointly measurable H: Comp(
Delta_M) x Omega ->M such that for all K in Comp(Delta_
M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) -->0, then for all p in Pi, p({omega: H(K_n,omega) -->H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's representation theorem.