The Equivalence of Two Extremum Problems

Let f 1, …, f k be linearly independent real functions on a space X, such that the range R of (f 1, …, f k) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the f i are continuous and X is a compact topological space. ) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = ε be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such that is defined. In all that follows we consider only probability measures ε which are in C.