Almost-sure variable-length source coding theorems for general sources

Source coding theorems for general sources are presented. For a source μ, which is assumed to be a probability measure on all strings of an infinite-length sequence with a finite alphabet, the notion of almost-sure sup entropy rate is defined; it is an extension of the Shannon entropy rate. When both an encoder and a decoder know that a sequence is generated by μ, the following two theorems can be proved: (1) in the almost-sure sense, there is no variable-rate source coding scheme whose coding rate is less than the almost-sure sup entropy rate of μ, and (2) in the almost-sure sense, there exists a variable-rate source coding scheme whose coding rate achieves the almost-sure sup entropy rate of μ