Supremal Generators and Their Applications

Abstract

A supremal generator of a set X of functions is a subset H of X such that each function from X is abstract convex with respect to H. In other words H is a supremal generator of X if each function from X can be represented as the upper envelope of a subset of H. As it turns out there exist very large sets with very small supremal generators. For example, the space of all lower semicontinuous functions defined on a segment of the real line has supremal generators, which are cones spanned by three functions only. If H is a supremal generator of a set of functions, then some properties of H allow one to study some properties of functions belonging to X. Properties of small supremal generators very often can be verified “manually”, that is by direct calculation. Thus the existence of small supremal generators is very helpful. In this chapter we first present a description of supremal generators for some sets of lower semicontinuous functions and then describe some small generators for these sets. We shall show that the description of supremal generators for the set of all lower semicontinuous functions is much easier than for proper subsets of this set.