I am following "Markov Chains and Stochastic stability" by Meyn and Tweedie. Throughout the book they use the convention that $\psi$ refers to a maximal irreducibility measure, and call the corresponding chain $\psi$-irreducible.
When a $\psi$-irreducible chain admits an invariant probability measure $\pi$, which satisfies
\begin{equation} \pi(A) = \int \pi(dx) P(x, A), \quad A \in \mathcal{B} (\mathsf{X}) \end{equation}
they call the chain positive. My question is this: Will we, for positive chains, have $\psi = \pi$?
I think this intuitively would make sense (since any set $A \in \mathcal{B}(\mathsf{X})$ with $\pi(A) > 0$ should clearly be returned to in finite time with positive probability), but I am not able to find any explicit statements of such a property in the book. Is it because the statement false, beacuse it is trivial, or beause I have missed something?