First class selectors for upper semi-continuous multifunctions

Abstract

By a multtfunction @: X + Y we mean a function defined on the space X and whose values are subsets of the space Y. A point x0 in X is a point of upper semi-continuity for @ provided x0 belongs to the interior of {x: Q(x) c G}, whenever G is an open set in Y containing @(x0); Q, is said to be upper semi-continuous if it is upper semi-continuous at ach point of X. In this case, {x: G(x) c G} is open in X, whenever G is open in Y, and {x:@(x)nF#@) is closed in X, w h enever F is closed in Y. Interchanging the roles of open and closed sets in this definition we get the concept of a lower semi-continuous multifunction. A selector for @ is a (single-valued) function cp: X -+ Y with q(x) E CD(X) for each x in X. In this paper we are primarily concerned with the problem of determin- ing the best possible selector cp for an upper semi-continuous multifunction @ defined on a metric space X and taking only non-empty values in a second metric space Y. Very elementary examples show that such multi- functions will in general fail to have continuous selectors: e.g., take 4P(x)={0}forO~x<~,~(~)=[O,1],and~(x)={1}for~<x~1.Hence an optimal selector in our case would be one that is the pointwise limit of a sequence of continuous functions, or, more generally, a first class Bore1 function (i.e., cp - ‘( G) is an 9$-set in X for each open set G c Y). Here we will prove