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This paper is concerned with two problems in the theory of source coding subject to a maximum average distortion constraint. The first problem involves the coding of a nonergodic discrete time source and the second involves coding for a class of ergodic discrete time sources. Coding theorems are given for both of these situations for very general source alphabets. The codes that are obtained here will, in general, be variable length codes so that average code rate and average distortion are the measures of performance.