Let $P$ be a transition matrix on the finite state space $S$. Show that every stationary distribution $\pi$ satisfies the following equation for all $A \subset S$:
$\sum_{i \in A} \sum_{j \in S \setminus A} \pi_{i}p(i,j) = \sum_{i \in S \setminus A} \sum_{j \in S} \pi_{i}p(i,j)$
Since $\pi$ is a stationary distribution, we know that all its components satisfy: $\sum \pi_{i}=1$. The same is true for the sum of all $p(i,j)$ in a row, as $P$ is a stochastic matrix. I started with:
$\sum_{i \in A} \sum_{j \in S \setminus A} \pi_{i}p(i,j) = \sum_{i \in A} \pi_{i} \sum_{j \in S \setminus A} p(i,j) = \sum_{i \in A} \pi_{i} (1-\sum_{j \in A} p(i,j))$
However, I don't know how to proceed. Does anyone of you have an idea?
Thank you very much for any thoughts or suggestions!