Characterization of convex -compact sets

Abstract

The class of μ-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class contains all compact sets as well as many noncompact sets widely used in applications. In this paper we give a characterization of a convex μ-compact set in terms of properties of functions defined on this set. Namely, we prove that the class of convex μ-compact sets can be characterized by continuity of the operation of convex closure of a function (= the double Fenchel transform) with respect to monotonic pointwise converging sequences of continuous bounded and of lower semicontinuous lower bounded functions. The properties of compact sets in the context of convex analysis have been studied by many authors (see [1, 2, 3] and the references therein). It is natural to ask about possible generalizations of the results proved for compact convex sets to noncompact sets. In [4] one such generalization concerning the particular class of sets called μ-compact sets is considered. In [4, 5] it is shown that for this class of sets, which includes all compact convex sets as well as some noncompact sets widely used in applications, many results of the Choquet theory [1] and of the Vesterstrom-O’Brien theory [2, 3] can be proved. In this paper we give a characterization of a convex μ-compact set in terms of properties of functions defined on this set. In what follows A is a bounded convex complete separable metrizable subset of some locally convex space. Let C(A) be the set of all continuous bounded functions on the set A and M(A) be the set of all Borel probability measures on the set A endowed with the weak convergence topology [6, Chapter II, §6]. Let cof and cof be the convex hull and the ∗e-mail:msh@mi.ras.ru This means that the topology on the set A is defined by a countable subset of the family of seminorms, generating the topology of the entire locally convex space, and this set is separable and complete in the metric generated by this subset of seminorms.