Summary. In recent years, convergence results for multivalued functions have been developed and used in several areas of applied mathematics: mathematical economics, optimal control, mechanics, etc. The aim of this note is to give a criterion of almost sure convergence for multivalued asymptotic martingales (amarts). For every separable Banach space B the fact that every L 1 -bounded B-valued martingale converges a.s. in norm to an integrable B-valued random variable (r.v.) is equivalent to the Radon-Nikodym property [6]. In this paper we solve the problem of a.s. convergence of multivalued amarts by giving a topological characterization. Preliminaries.Let X S be the set of all random elements (r.e.) defined on a probability space (Ω, A, P ) with values in a Polish (separable, complete metric) space (S, ), i.e. X S = {X : Ω → S; X −1 (B) ⊂ A}, where B = B S stands for the σ-field generated by the open subsets of S.Let P(S) denote the set of all probability measures defined on (S, B). The Lévy-Prokhorov metric on P(S) is defined as follows:where P X is the probability distribution of the r.e. X, A ε = {x : d(x, A) = inf y∈A (x, y) < ε}, andIt is known [1] that the convergence of a sequence of probability measures in the Lévy-Prokhorov metric and the weak convergence of this sequence coincide.

We consider different definitions of expectation of random elements taking values in metric spaces. All such definitions are valid also in Banach spaces and in this case the results coincide with the Bochner integral. There may exist an isometry between considered metric space and some Banach space and in this case one can use the Bochner integral instead of expectation in metric space. We give some conditions which ensure existence of such isometry, for two different definitions of expectation in metric space.

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