I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:

Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset \Omega^{\mathbb Z}$ be an irreducible Markov shift (i.e. an irreducible 1-step subshift of finite type). Denote by $(\Sigma,\mathcal B)$ the measure space with the usual product $\sigma$-algebra. Further, let $\mu$ be an ergodic, shift-invariant probability measure with full support on $(\Sigma,\mathcal B)$.

Question: Is $\mu$ necessarily a Markov measure?
**More precisely:** Is $\mu$ a measure that takes the value $p_{\omega_0} \prod_{t=1}^{\tau}\Pi_{\omega_{t},\omega_{t+1}}$ on the cylinder sets specified by the finite string $(\omega_0,\omega_1,\dots,\omega_\tau)$, where $\Pi_{i,j}$ are the elements of a compatible stochastic matrix and $p$ is its unique unity stochastic eigenvector?

It is quite easy to construct counter-examples if I drop certain assumptions. However, in the described set-up I could not find one. It feels like that question should have an easy answer, but somehow I don't get it.

Any help would be really appreciated!

Cheers, Bernhard