Q: Show that for $\mu (\partial \Omega)=0$ and $P(X_n \in C) \rightarrow 0$ as $n \rightarrow \infty$, $\Omega$ is bounded. Let $(X_n, n \ge 1)$ be a Markov chain whose state space is $\mathbb{R}$ equipped with Lebesgue measure. Let $\Omega \subset \mathbb{R}$ be a set with $\mu(\partial\Omega)=0$ and $P(X_n \in C) \rightarrow 0$ as $n \rightarrow \infty$ for every closed set $C$. Show that $\Omega$ is bounded. What I did was the following: Since $\mu(\partial\Omega)=0$, for every $\epsilon>0$ there exists an open ball $B$ around a point $a \in \Omega$ such that \$\mu(B \cap \partial\Omega) 6d1f23a050