Finitely additive measures in Markov general chain theory

The first part of the report gives a brief overview of the formation of a general theory of finitely additive measures from the 1930s to the present. There are two main established directions for their applications.
The second part is devoted to the use of finitely additive measures in the theory of probability, namely, in the theory of Markov chains. We consider the construction of the continuation of operators of general and topological Markov chains from the traditional space of countably additive measures to the space of finitely additive measures, which opens up new possibilities in the study of classical Markov chains. We emphasize that here we consider Markov chains with a traditional transition probability that is countably additive in the second argument.
Within the framework of this operator approach, the author's weak ergodic theorem is given for topological Markov chains that do not have invariant countably additive measures, but possess, like all Markov chains, invariant finite-additive measures (in this case, purely finitely additive). Examples are given.