Property of “correct exit” and one limit theorem for semi-Markov processes

A necessary and sufficient condition on weak convergence of a sequence or probability measures in the space D_{o, ∞]}(X) is formulated in terms of first exit times. The proof of necessity is based on continuity of first exit times and first exit points with respect to the Stone-Skorokhod metric on the set of functions that “correctly exit” from an open set Δ?X. A limit theorem for semi-Markov processes is proved as an application.