Type: Article
Publication Date: 1981-10-01
Citations: 22
DOI: https://doi.org/10.2140/pjm.1981.96.251
Let X be a compact metric space.If we identify each upper semicontinuons function / with its hypograph {(x, a): a ^ f(x)} in X 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 X R. Convergence with respect to this topology is in some respects analagous to convergence in measure.For example if {/"} is a sequence of continuous functions convergent to an u.s.c.limit /, then there exists a dense G δ set G such that for each x in G f(x) is a subsequential limit of {fn(%)} 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.1* Introduction* The analogies between Baire category and Lebesgue measure, so elegantly described by J. Oxtoby [7], have been from time to time the objects of study of some of the most eminent mathematicians of this century, including Banach, Ulam, Sierpinski, Erdos, and Kuratowski.Category duals of standard results in measure theory abound, although frequently the results may be stronger, weaker, or otherwise modified.Here is a typical example: MEASURE THEOREM.A set is in the o-algebra generated by the Borel sets and the sets of measure zero if and only if it can be represented as a F σ set plus a set of measure zero.CATEGORY DUAL.A set is in the σ-algebra generated by the Borel sets and the sets of first category if and only if it can be represented as a F σ set minus a set of first category.