On the optimum average distortion attainable by fixed-rate coding of a nonergodic source

A simple proof is given for the Gray-Davisson result on the ergodic decomposition of the optimum average distortion attainable by fixed-rate coding of a nonergodic source, for the special ease of a bounded distortion measure. A second proof is given for a more general class of distortion measures using a result of Ziv.