Stone-Weierstrass theorems for random functions

Abstract

We present several generalizations of the Stone-Weierstrass theorem concerning the approximation of continuous functions on a compact set by using functions from a subalgebra to the case of random functions and random variables in the space of continuous functions. The continuity of the random functions is allowed to be only with respect to a metric, hence including the case of stochastically continuous random functions. These results could be cornerstones for the general theory of approximation for random functions.