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

Optimal Nonstationary Reproduction Distribution for Nonanticipative RDF on Abstract Alphabets