Dimension of the space of invariant finitely additive measures of general Markov chains and their ergodic properties

Abstract

General Markov chains with a countably additive transition probability in arbitrary phase space are considered. Markov operators extend from the space of countably additive measures to the space of finitely additive measures. In the author's papers a theorem was earlier proved that if all invariant finitely additive measures of a Markov chain are countably additive, i.e. there are no invariant purely finitely additive measures, then their subspace is finite-dimensional and the Markov chain satisfies the Doob-Doeblin quasicompactness conditions. In the same paper, a partial inversion of this theorem was proved with the dimension "one". In this paper we prove the inversion of this assertion for any finite dimensionality, but under certain additional conditions. The ergodic consequences are given. Examples and methods for studying their asymptotics with the aid of invariant purely finitely additive measures are given.