# Is it unique when a irreducible and aperiodic markov chain on general space has an invariant measure?

Recently I'm reading Markov chains and Stochastics Stability(sencond edition 2009) written by Meyn and Tweedie. And in the proof of Theorem 10.4.5 on page 243, it says "if $\pi_m$ is invariant for the $m$-skeleton then by aperiodicity the measure is the unique invariant measure(up to multiples) for $\Phi^m$." is that right? If not, is there any counter-example?

Here is the Theorem 10.4.5:

Suppose that Markov Chain $\Phi$ is $\psi$-irreducible and aperiodic. Then, for each $m$, a measure $\pi$ is invariant for the $m$-skeleton if and only if it is invariant for $\Phi$.

• What's $m$-skeleton? – R W Dec 15 '13 at 5:16
• $\Phi_0, \Phi_m, \Phi_{2m}, \Phi_{3m}, ...$ picked out from the original chain – Wieshawn Dec 15 '13 at 6:08