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.