 Constant-drift hitting-time identity

For a finite Markov chain, let $T$ be the first hitting time of $D$, assume $T<\infty$ almost surely, $f=0$ on $D$ and $Pf=f-\delta$ on its complement, with $\delta >0$. The stopped martingale $f(X_{n\wedge T})+\delta (n\wedge T)$ has mean $f(x)$. Boundedness of $f$ and bounded convergence theorem give $\mathbb{E}f(X_{n\wedge T})\to 0$; monotone convergence then gives the stated finite expected hitting time. No drift condition is required after entry into $D$.