Let $(X_i)$ be an irreducible Markov chain on a finite set $E$ with transition matrix $P$ and invariant distribution $\pi$. Let $T$ be the shift on $E^{\mathbb{N}}$, $$ T: E^{\mathbb{N}}\rightarrow E^{\mathbb{N}} $$ $$ (x_0,x_1,x_2,...)\mapsto (x_1,x_2,x_3,...) $$
The french wiki on Markov chains says here that the following dynamical system $(E,\Pi,T)$ is ergodic ($E$ is endowed with the product topology and $\Pi$ is the "product" measure). I see why it is measure preserving, can anyone help me to show that for a measurable set $A$, $T^{-1}(A)\subset A$ implies $\Pi(A)=1$ or $0$?
Thank you