Skorohod’s representation theorem for sets of probabilities

Abstract

We characterize sets of probabilities, Π \boldsymbol {\Pi } , on a measure space ( Ω , F ) (\Omega ,\mathcal {F}) , with the following representation property: for every measurable set of Borel probabilities, A A , on a complete separable metric space, ( M , d ) (M,d) , there exists a measurable X : Ω → M X:\Omega \rightarrow M with A = { X ( P ) : P ∈ Π } A = \{X(P): P \in \boldsymbol {\Pi }\} . If Π \boldsymbol {\Pi } has this representation property, then: if K n → K 0 K_n \rightarrow K_0 is a sequence of compact sets of probabilities on M M , there exists a sequence of measurable functions, X n : Ω → M X_n:\Omega \rightarrow M such that X n ( Π ) ≡ K n X_n(\boldsymbol {\Pi }) \equiv K_n and for all P ∈ Π P \in \boldsymbol {\Pi } , P ( { ω : X n ( ω ) → X 0 ( ω ) } ) = 1 P(\{\omega : X_n(\omega ) \rightarrow X_0(\omega )\}) = 1 ; if the K n K_n are convex as well as compact, there exists a jointly measurable ( K , ω ) ↦ H ( K , ω ) (K,\omega ) \mapsto H(K,\omega ) such that H ( K n , Π ) ≡ K n H(K_n,\boldsymbol {\Pi }) \equiv K_n and for all P ∈ Π P \in \boldsymbol {\Pi } , P ( { ω : H ( K n , ω ) → H ( K 0 , ω ) } ) = 1 P(\{\omega : H(K_n,\omega ) \rightarrow H(K_0,\omega )\}) = 1 .