# showing existence of invariant measure

This is somehow, related to my last doubt(question), Let $$X_n(x_0)$$ be a (weak) Feller Markov chain on $$[0,1]$$, starting from $$x_0\in \mathbb [0,1]$$. I am given that $$\lim\limits_{n\to \infty}\sum\limits_{k=0}^{n-1}\text{prob}(X_k(x_0)\in K)>0$$, where $$K$$ is a compact subset of $$[0,1]$$ with $$\mu(K)=0$$, where $$\mu$$ is Lebesgue measure. Now could any-one tell me how to show that there is an invariant measure(weak)?[Hint: Portmanteau theorem]. Thanks so much for any little help to proceed!

• Is this a homework question? – Anthony Quas Nov 28 at 16:47