This paper introduces a general framework for dealing with the subject of rate distortion or source coding with fidelity criterion, in Polish spaces. Previous well known results for abstract alphabets are extended to the more general case in which marginal measures on the reproduction space may not be absolutely continuous with respect to the optimal marginal measure. In this situation, the chain rule for Radon-Nikodym derivatives does not hold and hence standard techniques do not apply. This problem is resolved by developing a variational approach on the space of measure valued functions. It is also shown that the compactness assumption on the reproduction space can be relaxed. Moreover, the question of existence of a solution to the implicit equation of optimal distribution is addressed by proving a fixed point theorem. In the discrete alphabet case, the existence of such a solution follows from the Blahut algorithm. However in the abstract alphabet case, more analysis is needed. This is formulated as a fixed point problem on a suitable locally convex topological vector space. Existence of solution of the original problem is proved by establishing the existence of a fixed point.
Rate distortion theory, Banach spaces, Polish spaces, Finitely additive measures, Lower semi-continuity, Fixed point theorem
