Stone-Weierstrass theorems for random functions

Abstract

We present several generalizations of Stone-Weierstrass theorem about the approximation of continuous functions on a compact set by function 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 of a general approximation theory for random functions.