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.