Geodesic
The final $\sigma$-algebra of an inhomogeneous Markov chain with a finite number of states
Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 295-302.

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It is proved that the final $\sigma$-algebra in the case of an inhomogeneous Markov chain with a finite number of states $n$ is generated by a finite number ($\leqslant n$) of atoms. The atoms are characterized from the point of view of the behavior of trajectories of the chain. Sufficient conditions are given (in the case of a countable number of states) that there should exist an unique atom at infinity. 
@article{MZM_1972_12_3_a10,
     author = {D. V. Senchenko},
     title = {The final $\sigma$-algebra of an inhomogeneous {Markov} chain with a finite number of states},
     journal = {Matemati\v{c}eskie zametki},
     pages = {295--302},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a10/}
}
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D. V. Senchenko. The final $\sigma$-algebra of an inhomogeneous Markov chain with a finite number of states. Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 295-302. http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a10/