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 $\Phi$.