On the filtering problem for stationary random Z2-fields

It is shown that whenever a stationary random field (Z(n,m))(n,m is an element of z) is given by a Borel function f : R-z x R-z -> R of two stationary processes (X-n)(n is an element of z) and (Y-m)(m is an element of z), i.e., (Z(n,m)) = (f((Xn+kappa)(kappa is an element of z), (Ym+l)(l is an element of z))) then under a mild first coordinate univalence assumption on f, the process (X-n)(n is an element of z) is measurable with respect to (Z(n,m))(n),(m is an element of z) whenever the process (Y-m)(m is an element of z) is ergodic. The notion of universal filtering property of an ergodic stationary process is introduced, and then using ergodic theory methods it is shown that an ergodic stationary process has this property if and only if the centralizer of the dynamical system canonically associated with the process does not contain a nontrivial compact subgroup.