# condition for positive singular part

Could anyone give me an example of an invariant measure with a positive singular part? I need to prove that a Feller Markov chain on $$\mathbb R$$ has an invariant measure with positive singular part if and only if there exists a closed and bounded set $$B$$ with $$\mu(B)=0$$ where $$\mu$$ is Lebesgue measure, and $$\frac{1}{t}\sum\limits_{j=0}^{t-1} P(X_j(x_0)\in B)>0$$ as $$t\to \infty$$. Thanks so much for your help. where $$X_j(x_0)$$ is a Markov chain whose starting point is $$x_0$$. (Hint: Portmantau Theorem), I am self-studying measure and probability theory! Thanks for some ideas or reference. Is it a good place to learn the Portmantau theorem?