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

TL;DR

This paper establishes that for general Markov chains, ergodicity conditions like Doeblin's are equivalent to the absence of purely finitely additive invariant measures, linking ergodic behavior to measure-theoretic properties.

Contribution

It proves the equivalence of classical ergodicity conditions with measure-theoretic properties involving finitely additive measures in general Markov chains.

Findings

Doeblin condition (D) is equivalent to all invariant finitely additive measures being countably additive.

Conditions (D) and (*) are equivalent to the finite-dimensionality of the set of invariant finitely additive measures.

Ergodic theorems are established for general Markov chains under these conditions.

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.