The final $\sigma$-algebra of an inhomogeneous Markov chain with a finite number of states

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.