  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.
