Upper semicontinuous functions and the stone approximation theorem

Abstract

In convex function theory it has long been recognized as useful to identify a convex function with its epigraph, the convex set of points on or above its graph. Similarly, a concave function is identified with its hypograph, the convex set of points on or below its graph. Analysis is then performed in the product space. We present two standard examples. First, the interior of the epigraph of a convex function f consists of those points (x, a) for which x lies in the interior of the domain off and a > f(x). As a result f is upper semicontinuous on the interior of its domain. Second, a closed convex set is the intersection of the closed half spaces that contain it. As a result, a lower semicontinuous convex function is the pointwise supremum of the affine functions that it majorizes. Such techniques have enjoyed only limited popularity in other branches of real analysis; that is, the topology and/or linear structure of the graph, epigraph, or hypograph of a real valued function are rarely used to define the fundamental concepts of analysis or to prove theorems. Their role has been essentially descriptive. It is the purpose of this paper to “perform analysis in the product” to gain a new understanding of the notion of uniform approximation. Our basic tool will be the Hausdorff metric on closed subsets of the product. Using this metric we present a generalization of the Stone Approximation Theorem to the space of upper semicontinuous functions defined on a compact metric space. In the process we extend Dini’s theorem, characterizing those sequences of upper semicontinuous functions convergent pointwise from above to a continuous function that converge uniformly. Finally, since the topology on the continuous functions on a compact metric space induced by the Chebyshev norm coincides with the one induced by the Hausdorff metric when restricted to their graphs, we obtain a different view of equicontinuity and its place in the Arzela-Ascoli theorem.