Given a stochastic matrix $p(x,y)$ on an discrete Polish space $S$. The associated Markov operator $P$ is a contraction on $\ell_\infty(S)$. 1. If for each finite subset $K$ of $S$ and each $\e > 0$ there is another finite subset $E$ of $S$ such that for all $x\notin E$: $P(x,K) < \e$, then $P$ is a contraction on $c_0(S)$ and thus $P^*$ is a contraction on $\ell_1(S)$ (the dual of $c_0(S)$ is isometrically isomorphic to $\ell_1(S)$). 2. A (signed) measure $\mu$ on $S$ is just a vector $\mu\in\ell_1(S)$. Verify that if $P$ is a contraction on $c_0(S)$ then $\mu$ is invariant iff $P^*\mu=\mu$, i.e. $$ \forall x\in S:\quad \sum_{y\in S}\mu(y)p(y,x)=\mu(x)~. $$ Give an example of an infinite stochastic matrix such that $P$ doesn't map $c_0(S)$ into $c_0(S)$.