Gluing lemmas and Skorohod representations

Abstract

Let (X; ɛ), (Y,F) and (Z,G) be measurable spaces. Suppose we are given two probability measures γ and τ, with γ defined on (X × Y, E ⊗ F) and τ on (X× Z, E ⊗ G). Conditions for the existence of random variables X,Y, Z, defined on the same probability space (Ω,A,P) and satisfying (X; Y) ∼ γ and (X;Z) ∼ τ, are given. The probability P may be finitely additive or -additive. As an application, a version of Skorohod representation theorem is proved. Such a version does not require separability of the limit probability law, and answers (in a finitely additive setting) a question raised in [2] and [4].