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). Similarly, by the Hahn-Banach theorem, if A separates probability measures, A is dense in C (X). We show 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.