$\omega$-regular properties of a 2-state Markov Chain
I am not sure whether $A_f$ is an $\omega$-regular event, but for any $\delta&gt;0$ we can pick some $\omega$-regular $A_\delta$ such that $|\mathsf P(A_\delta) - \mathsf P_\delta(A_\delta)| \geq 1-\delta$. This follows from the fact that $\omega$-regular events form an algebra that generates a $\sigma$-algebra where $A_f$ does belong to.