  The class of subsets of locally convex spaces called $\mu$-compact sets is
considered. This class contains all compact sets as well as several noncompact
sets widely used in applications. It is shown that many results well known for
compact sets can be generalized to $\mu$-compact sets. Several examples are
considered.
  The main result of the paper is a generalization to $\mu$-compact convex sets
of the Vesterstrom-O'Brien theorem showing equivalence of the particular
properties of a compact convex set (s.t. openness of the mixture map, openness
of the barycenter map and of its restriction to maximal measures, continuity of
a convex hull of any continuous function, continuity of a convex hull of any
concave continuous function). It is shown that the Vesterstrom-O'Brien theorem
does not hold for pointwise $\mu$-compact convex sets defined by the slight
relaxing of the $\mu$-compactness condition. Applications of the obtained
results to quantum information theory are considered.
