  In this paper, we consider general Markov chains (MC), specified by the
transition probability (kernel) $ P (x, E) $, finitely additive in the second
argument. Such MC are studied within the framework of the functional operator
treatment. The state space (phase space) of the MC $ X $ has any cardinality,
and the sigma-algebra $ \Sigma $ is discrete, i.e. is the set of all subsets in
$ X $. This construction of the phase space $ (X, \Sigma) $ allows us to
decompose the Markov kernel $ P (x, E) $ into the sum of two components -
countably additive and purely finitely additive in the second argument and
measurable in the first argument. It is shown that the countably additive
kernel is atomic. Some properties of Markov operators with a purely finitely
additive kernel and their invariant measures are studied. A class of combined
finitely additive MC and two of its subclasses are introduced, and some
properties of their invariant measures are proved. Some asymptotic regularities
of such MC are revealed.
