Vol. 96, No. 2, 1981

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A natural topology for upper semicontinuous functions and a Baire category dual for convergence in measure

Gerald A. Beer

Vol. 96 (1981), No. 2, 251–263

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.

Mathematical Subject Classification 2000
Primary: 54C08
Secondary: 28A20, 54C35
Received: 26 June 1980
Revised: 2 September 1980
Published: 1 October 1981
Gerald A. Beer