Skorohod’s representation theorem for sets of probabilities

Abstract:

We characterize sets of probabilities, $\boldsymbol {\Pi }$, on a measure space $(\Omega ,\mathcal {F})$, with the following representation property: for every measurable set of Borel probabilities, $A$, on a complete separable metric space, $(M,d)$, there exists a measurable $X:\Omega \rightarrow M$ with $A = \{X(P): P \in \boldsymbol {\Pi }\}$. If $\boldsymbol {\Pi }$ has this representation property, then: if $K_n \rightarrow K_0$ is a sequence of compact sets of probabilities on $M$, there exists a sequence of measurable functions, $X_n:\Omega \rightarrow M$ such that $X_n(\boldsymbol {\Pi }) \equiv K_n$ and for all $P \in \boldsymbol {\Pi }$, $P(\{\omega : X_n(\omega ) \rightarrow X_0(\omega )\}) = 1$; if the $K_n$ are convex as well as compact, there exists a jointly measurable $(K,\omega ) \mapsto H(K,\omega )$ such that $H(K_n,\boldsymbol {\Pi }) \equiv K_n$ and for all $P \in \boldsymbol {\Pi }$, $P(\{\omega : H(K_n,\omega ) \rightarrow H(K_0,\omega )\}) = 1$.