Weak ergodic theorem for Markov chains without invariant countably additive measures

Abstract: In this paper, we study Markov chains (MC) on topological spaces within the framework of the operator approach. We extend the Markov operator from the space of countably additive measures to the space of finitely additive measures. Cesaro means for a Markov sequence of measures and their asymptotic behavior in the weak topology are considered. It is proved ergodic theorem that in order for the Cesaro means to converge weakly to some bounded regular finitely additive (or countably additive) measure it is necessary and sufficient that all invariant finitely additive measures are not separable from the limit measure in the weak topology. Moreover, the limit measure may not be invariant for a MC, and may not be countably additive. The corresponding example is given and studied in detail.