A Stone-Weierstrass theorem without closure under suprema

Abstract:

For a compact metric space $X$, consider a linear subspace $A$ of $C\left ( X \right )$ 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 $C\left ( X \right )$. By the Hahn-Banach Theorem, if $A$ separates probability measures, $A$ is dense in $C\left ( X \right )$. It is shown that if $A$ separates points from probability measures, then the closure of $A$ under minima is dense in $C\left ( X \right )$. This theorem has applications in economic theory.