General Markov Chains: Dimension of the Space of Invariant Finitely Additive Measures and Their Ergodicity—Problematic Examples

This study considers general Markov chains (MCs) with discrete time in an arbitrary phase space. The transition function of the MC generates two operators: T, which acts on the space of measurable functions, and A, which acts on the space of bounded countably additive measures. The operator $T^{*}$, which is adjoint to T and acts on the space of finitely additive measures, is also constructed. A number of theorems are proved for the operator $T^{*}$, including the ergodic theorem. Under certain conditions it is proved that if the MC has a finite number of basic invariant finitely additive measures then all of them are countably additive and the MC is quasi-compact. We demonstrate a methodology that applies finitely additive measures for the analysis of MCs, using examples with detailed proofs of their non-simple properties. Some of these proofs in the examples are more complicated than the proofs in our theorems.