  The relation between nonanticipative Rate Distortion Function (RDF) and
filtering theory is discussed on abstract spaces. The relation is established
by imposing a realizability constraint on the reconstruction conditional
distribution of the classical RDF. Existence of the extremum solution of the
nonanticipative RDF is shown using weak$^*$-convergence on appropriate
topology. The extremum reconstruction conditional distribution is derived in
closed form, for the case of stationary processes. The realization of the
reconstruction conditional distribution which achieves the infimum of the
nonanticipative RDF is described. Finally, an example is presented to
illustrate the concepts.
