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General Markov chains with a countably additive transition probability inarbitrary phase space are considered. Markov operators extend from the space ofcountably additive measures to the space of finitely additive measures. In theauthor's papers a theorem was earlier proved that if all invariant finitelyadditive measures of a Markov chain are countably additive, i.e. there are noinvariant purely finitely additive measures, then their subspace isfinite-dimensional and the Markov chain satisfies the Doob-Doeblinquasicompactness conditions. In the same paper, a partial inversion of thistheorem was proved with the dimension "one". In this paper we prove theinversion of this assertion for any finite dimensionality, but under certainadditional conditions. The ergodic consequences are given. Examples and methodsfor studying their asymptotics with the aid of invariant purely finitelyadditive measures are given.

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