Given a Markov chain with state space $\Omega$ and transition matrix $P$, and $A\subset \Omega$, define function$f(x)=\Bbb E _x(\tau_A)$, where $\tau_A$ is the "stopping time into set $A$", meaning the first time the Markov chain reaches a state in $A$.
It has been shown that $f(x)=0$ for $x\in A$, and $f\left( x \right) = 1 + \mathop \sum \limits_{y \in {\rm{\Omega }}} P\left( {x,y} \right)f\left( y \right)$ for $x\notin A$.
The question is to show $f$ is uniquely determined.
I have been puzzled by this problem for a while and have not get a clue. Hope someone can help. Thank you!
The following is part of the textbook that might be related to this problem, which shows the uniqueness of the stationary distribution.
The solution in image. I will find some time to type in latex later.
The solution uses the result of a previous problem.