  The class of $\mu$-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
$\mu$-compact set in terms of properties of functions defined on this set.
Namely, we prove that the class of convex $\mu$-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.
