Purification of measure-valued maps

Abstract

Given a measurable mapping $f$ from a nonatomic Loeb probability space $(T,\mathcal{T},P)$ to the space of Borel probability measures on a compact metric space $A$, we show the existence of a measurable mapping $g$ from $(T,\mathcal{T},P)$ to $A$ itself such that $f$ and $g$ yield the same values for the integrals associated with a countable class of functions on $T\times A$. A corollary generalizes the classical result of Dvoretzky-Wald-Wolfowitz on purification of measure-valued maps with respect to a finite target space; the generalization holds when the domain is a nonatomic, vector-valued Loeb measure space and the target is a complete, separable metric space. A counterexample shows that the generalized result fails even for simple cases when the restriction of Loeb measures is removed. As an application, we obtain a strong purification for every mixed strategy profile in finite-player games with compact action spaces and diffuse and conditionally independent information.