Let X be a compact metric
space. If we identify each upper semicontinuons function f with its hypograph
{(x,α) : α ≦ f(x)} in X × R, then the set UC(X) of all u.s.c. functions can be
viewed as a metric subspace of the hyperspace of X × R. Convergence with respect
to this topology is in some respects analagous to convergence in measure.
For example if {fn} is a sequence of continuous functions convergent to an
u.s.c. limit f, then there exists a dense Gδ set G such that for each x in G f(x) is
a subsequential limit of {fn(x)}. Integral convergence theorems are also
presented. However, the main results are as follows: (a) a characterization of this
topology on UC(X) in terms of the monotone functionals on C(X) that are
u.s.c. with respect to the uniform metric (b) several characterizations of
sublattices of UC(X) from which UC(X) is retrievable via pointwise limits of
monotone decreasing sequences, e.g., C(X) or the sublattice of u.s.c. step
functions.
