I'm trying to find the proof for a claim which seems to be a stronger version of the convergence theorem for finite Markov chains. I would appreciate the help in that regard.
The claim is as follows:
Let $\left\{ x_{t}\right\}$ be a Markov chain over a finite state space $\chi$ with an aperiodic transition matrix P.
for every essential communicating class $C$, define $h_{C}$ as the only harmonic function that satisfies
$h_{C}\left(y\right)=1\iff y\in C$
$h_{C}\left(y\right)=0\iff y\notin C$
Let $\mu_{0}^{x}$ a distribution that satisfies for all $y\in\chi$ $\mu_{0}^{x}\left[y\right]=1\iff y=x$.
Then
$$ \mu_{0}^{x}P^{t}\rightarrow\sum_{C\text{ essential communicating class }}h_{C}\left(x\right)\pi_{C} $$