Several properties of operational and information theoretic distortion rate functions and of mutual information rate are developed for asymptotically mean stationary (ams) processes with standard alphabets and used to prove a block source coding theorem for ergodic ams sources. In the development we prove an ergodic decomposition of the mutual information rate of ams sources, we show that the operational distortion rate function of an ams source equals that of its stationary mean, and we provide a new and direct proof of the equality of the ergodic and stationary process distortion rate functions. The latter result yields a proof of the block source coding theorem not requiring the Nedoma decomposition and complicated code construction used by Gallager and Berger.