Cycles in Spaces of Finitely Additive Measures of General Markov Chains

Abstract

AbstractGeneral Markov chains in an arbitrary phase space are considered in the framework of the operator treatment. Markov operators continue from the space of countably additive measures to the space of finitely additive measures. Cycles of measures generated by the corresponding operator are constructed, and algebraic operations on them are introduced. One of the main results obtained is that any cycle of finitely additive measures can be uniquely decomposed into the coordinate-wise sum of a cycle of countably additive measures and a cycle of purely finitely additive measures. A theorem is proved (under certain conditions) that if a finitely additive cycle of a Markov chain is unique, then it is countably additive.KeywordsGeneral Markov chainsMarkov operatorsFinitely additive measuresCycles of measuresDecomposition of cycles