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General Markov chains in an arbitrary phase space are considered in theframework of the operator treatment. Markov operators continue from the spaceof countably additive measures to the space of finitely additive measures.Cycles of measures generated by the corresponding operator are constructed, andalgebraic operations on them are introduced. One of the main results obtainedis that any cycle of finitely additive measures can be uniquely decomposed intothe coordinate-wise sum of a cycle of countably additive measures and a cycleof purely finitely additive measures.We have proved theorems on the conditionsand consequences of consistency cycles of measures with cycles of sets ofstates of General Markov chains. A theorem is proved (under certain conditions)that if a finitely additive cycle of a Markov chain is unique, then it iscountably additive.

General Markov Chains: Cycles of Finitely Additive Measures and Classical Cycles of States