Convergence of measurable random functions

Using the theorem of Fréchet and Kolmogorov about compact subsets of the space L_p[0, 1] and Prohorov's theorem about the convergence of measures defined on a complete metric space we proved in [2] the following theorem: Let ξ₁(t),…, ξ_n,(/)>… and ξ (t) be measurable random processes (0 ≤ t ≤ 1) and suppose that there exist numbers C and p ≥ 1 such that E|ξ_n(t)^p| ≤ C for all n and t. If E|ξ_n(t)^p|→ £|£(0lp for all /and if for any finite set {t₁,…, t_k] c [0, 1] the joint distribution of £„(f,),…, £„(tk) converges to the joint distribution of ξ (t_n),…, ξ (t_k), then the distribution of F(ξ_n) converges to the distribution of F(ξ) for any continuous functional F on L_P,[0, 1]. In this paper this theorem is generalized to random measurable functions. The results of the present paper are related to the results of [1], [3], [4].