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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?

[http://theanalysisofdata.com/probability/8_5.html]

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