A Stone-Weierstrass Theorem without closure under suprema

Abstract

For a compact metric space X, consider a linear subspace A of C(X) containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if A separates points, then the closure of A under both minima and maxima is dense in C(X). By the Hahn-Banach Theorem, if A separates probability measures, A is dense in C(X). It is shown that if A separates points from probability measures, then the closure of A under minima is dense in C(X). This theorem has applications in economic theory.