The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, *Ergodic Theorems*, section 2.1. At the beginning of the same book (end of p.4), Krengel mentions that such a result is *not* true for $L_\infty$.

More precisely, he claims that letting $\Omega=[0,1[$, endowed with the Lebesgue measure, and considering the measure-preserving transformation $\tau\omega:=\omega+\alpha$ (mod 1) where $\alpha$ is irrational, one can find a suitable *(highly discontinuous)* function $f\in L_\infty(\Omega)$ such that $A_nf:=(f+f\circ\tau+f\circ\tau^2+\ldots+f\circ\tau^{n-1})/n$ does *not* converge to $\bar f:=\int_0^1f$ in the $L_\infty(\Omega)$ norm. He leaves this fact as an exercise.

Does anyone know how to show this?