Thank you for the answer, on the "language of Markov chains" the statement seems really obvious. However, the problem was to find an explanation on the "language of shifts".

To Ilya: by Markov measure I mean exactly the concept defined here: en.wikipedia.org/wiki/Subshift_of_finite_type

Yes, I assume that $(A^\mathbb{N}, \sigma, m)$ is a measure-preserving dynamical system, namely shift of finite type with Markov Measure (sometimes it is called Markov shift).

to R W: yes, exactly

To Ilya: I consider a shift system $(A^\mathbb{N}, \sigma, m)$ over some finite "alphabet" $A$, therefore "open" means open in product topology

To HW: the context is not easy to describe and it does not seem to be helpful - this is a small separate problem. The global problem is to show that some special partition has good property (similar to the properties of cylinder sets partition). "Doesn't it follow immediately from the case of $P$ a cylinder set" - this would be great, but I'm not sure that the statement of interest obviously follows from this case (may be I just do not understand some simple thing).