Narrow Convergence in Spaces of Set-Valued Measures

Abstract

Summary. We prove an analogue of Topsoe's criterion for relative compactness of a family of probability measures which are regular with respect to a family sets. We consider measures whose values are compact convex sets in a locally convex linear topological space. Introduction. Let T be an abstract set, K a family of subsets of T, and (E;F) a dual pair of real vector spaces, with E endowed with the weak topology (E;F). Let cc(E;F) be the set of all convex compact non-empty subsets of E, and f M+(T;K;cc(E;F)) the set of K-inner regular positive set-valued measures defined on a -fieldB of subsets of T and with values in cc(E;F). We denote by M+(T;K) the set ofK-inner regular non-negative measures defined on B provided with the topology of weak convergence. Prokhorov (11) has proved that if T is a Polish space andB the set of Borel subsets, then the relatively compact subsets of M+(T;K) are precisely the tight ones. But this result is not valid for all topological space (see e.g. (5), (10), (18)). In (16) Topsoe has characterized the relatively compact subsets of M+(T;K) in general situations. Before and after Topsoe's paper there were others (e.g. (1), (3), (18), (5)-(10)). In this paper we generalize to the space f