A Stone-weierstrass Theorem without Closure under Suprema

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. The classical Stone-Weierstrass Theorem states that, if a linear space A of real valued functions defined on a compact metric space X contains the constant functions, is closed under minima and maxima, and separates points, then A is dense in C(X). The purpose of this paper is to provide an alternative structure for sets closed under minima alone, which generates the same result. The theorem fits between-the Stone-Weierstrass Theorem and a corollary to the Hahn-Banach Theorem. Let X be a compact metric space, with metric p, and A the set of probability measures (regular unitary measures) on X. Let 6X represent the point mass measures: