Ergodicity conditions for general Markov chains in terms of invariant finitely additive measures

Abstract: We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment generates a conjugate pair of linear Markov operators in the Banach space of measurable bounded functions and in the Banach space of bounded finite additive measures. It is proved that the well-known Doeblin condition $ (D) $ of ergodicity (quasi-compactness) of the Markov chain is equivalent to the condition $ () $: all finitely additive invariant measures of the Markov operator are countably additive i.e. there are no invariant purely finitely additive measures. Under some assumptions, it is proved that the conditions $ (D) $ and $ () $ are also equivalent to the condition $ (**) $: the set of invariant finitely additive measures of a Markov operator is finite-dimensional. Ergodic theorems are given.