Lossy universal source coding for individual sequences

Under weak restriction on the distortion measure the continuity of the. single letter distortion measure is established with respect to a suitable distance measure on the sequence space of elements drawn from the representation alphabet. This in turn translates into properties of the distortion rate function. Using subtleties about the behavior of the sequence distortion measure we develop a proof of convexity of the distortion rate function in full generality. A generalization of the optimality of Lempel Ziv algorithm with regard to finite state codes and stationary and ergodic sources with finite alphabet is presented for a more general class of universal codes applied to individual sequences.