A Strong Version of the Skorohod Representation Theorem

For each $$n\ge 0$$ n ≥ 0 , let $$\mu _n$$ μ n be a tight probability measure on the Borel $$\sigma $$ σ -field of a metric space S. Let $$(T,{\mathcal {C}})$$ ( T , C ) be a measurable space such that the diagonal $$\bigl \{(t,t):t\in T\bigr \}$$ { ( t , t ) : t ∈ T } belongs to $${\mathcal {C}}\otimes {\mathcal {C}}$$ C ⊗ C . Fix a measurable function $$g:S\rightarrow T$$ g : S → T and suppose $$\mu _n=\mu _0$$ μ n = μ 0 on $$g^{-1}({\mathcal {C}})$$ g - 1 ( C ) for all $$n\ge 0$$ n ≥ 0 . Necessary and sufficient conditions for the existence of S-valued random variables $$X_n$$ X n , defined on the same probability space and satisfying $$\begin{aligned} X_n\overset{\text {a.s.}}{\longrightarrow }X_0,\quad X_n\sim \mu _n\,\text { and } \,g(X_n)=g(X_0)\,\text { for all }n\ge 0, \end{aligned}$$ X n ⟶ a.s. X 0 , X n ∼ μ n and g ( X n ) = g ( X 0 ) for all n ≥ 0 , are given. Such conditions are then applied to several examples. The tightness condition on $$\mu _0$$ μ 0 can be dropped at the price of some assumptions on S and g.