Unbounded mappings and weak convergence of measures

Weak convergence of finite Borel measures in a completely regular topological space X is defined by means of the class of bounded and continuous functions f:X→R. We give conditions equivalent to weak convergence of finite measures in terms of some classes of unbounded, continuous and semicontinuous real-valued functions on X. For this purpose we introduce the notion of almost uniformly integrable mappings with respect to a directed family of measures. The obtained results may be treated as an extension of the Alexandroff theorem, also known as portmanteau theorem.