A Generalized Typicality for Abstract Alphabets

Abstract: A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including conditional typicality lemma, joint typicality lemma, and packing and covering lemmas, which are fundamental tools for deriving many inner bounds of various multi-terminal coding problems, are obtained in terms of the proposed notion. This enables us to directly generalize lots of results on finite alphabet problems to general problems involving abstract alphabets, without any complicated additional arguments. For instance, quantization procedure is no longer necessary to achieve such generalizations. Another fundamental lemma, Markov lemma, is also obtained but its scope of application is quite limited compared to others. Yet, an alternative theory of typical sets for Gaussian measures, free from this limitation, is also developed. Some remarks on a possibility to generalize the proposed notion for sources with memory are also given.