Quasi-Feller Extensions of Markov Chains and Existence of Dual Chains

The author presents a revised and detailed version of his theorem on the existence of Feller's extensions of Markov chains; to this end the broader notion of quasi-Feller extension is used. The existence of Markov chains dual to the chains with Borel space of states is derived from this result. Chains irreducible in the Orey sense are studied in most detail. For example, we prove that for such chains the quasi-Feller extension can be chosen recurrent or Liouville if the original chains possess these properties.