Source coding theorems without the ergodic assumption

Source coding theorems are proved for discrete-time stationary processes subject to a fidelity criterion. The alphabet of the process is assumed to be a separable metric space, but the process is not assumed to be ergodic. When the process is not ergodic, the minimum average distortion for a fixed-rate code is not given by the distortion-rate function of the source as usually defined. It is given instead by a weighted average of the distortion-rate functions of ergodic subsources comprising the ergodic decomposition of the source. Potential applications to universal source coding with a fidelity criterion are discussed.