  In this paper we introduce a definition for nonanticipative Rate Distortion
Function (RDF) on abstract alphabets, and we invoke weak convergence of
probability measures to show various of its properties, such as, existence of
the optimal reproduction conditional distribution, compactness of the fidelity
set, lower semicontinuity of the RDF functional, etc. Further, we derive the
closed form expression of the optimal nonstationary reproduction distribution.
This expression is computed recursively backward in time. Throughout the paper
we point out an operational meaning of the nonanticipative RDF by recalling the
coding theorem derive in \cite{tatikonda2000}, and we state relations to
Gorbunov-Pinsker's nonanticipatory $\epsilon-$entropy \cite{gorbunov-pinsker}.
