On convergences of probability measures in different Prohorov-type metrics (Q1970831)

The author investigates a `uniform approximation property' of sequences of probability measures which was used in a previous paper of the author to establish a conditional convergence theorem of the form \(E(Z\mid X_n) \to E(Z\mid X)\) for some Skorokhod representation \(X_n\) of \(P_n\), \(X\) of \(P\). It is shown that the uniform approximation property together with convergence in the Prokhorov metric (i.e. weak convergence) implies convergence in the minimal \(L^\infty\)-metric (denoted \(\pi_\infty\)). Under some weaker form of uniform approximation equivalence holds.