Convergence of measures and random processes

This article is concerned with general theorems on the so-called $\mathscr F$-convergence of measures on an arbitrary measurable space. This is defined as weak convergence of measures for a given class $\mathscr F$ of functional. It seems that such a convergence concept is the most natural one in the study of limit theorems for random processes. The method used reduces the problems to the study of convergence of measures and charges in the topological spaces of A. D. Aleksandrov (or $\sigma$-spaces, see [1]). Such an approach was adopted in [5] and made it possible to establish in a unified way a number of new results as well as almost all the known limit theorems on the convergence of measures. The convergence conditions obtained here for measures in ordinary topological spaces are very similar to the conditions that had been introduced earlier by Prokhorov [27], Le Cam [24], Topsoe [31], Varadarajan [32], Dudley [10], Erokhin [14] and other authors. Without a doubt, these authors have fundamentally influenced both the contents and the structure of this article.
The major part of the work is devoted to applications of the results to concrete problems in random processes. At the end we discuss other possible approaches to the study of convergence of processes: the so-called method of a single probability space of Skorokhod [29] and the “approximative” method expounded in [4].
The sections marked by an asterisk are somewhat secondary to the basic exposition and may be omitted at a first rapid reading. The symbol $\blacksquare$ denotes the end of a proof.