Let $\tau:\Omega\to \Omega$ be a measure-preserving transformation with $\mu(\Omega)<\infty$. Define $T:L_p(\Omega)\to L_p(\Omega)$ as $Tf:=f\circ \tau$. I want to prove that for all $1\leq p<\infty$, given $f\in L_p$ there exists $\bar{f}\in L_p$ such that $\|\bar{f}\|_p\leq \|f\|_p,$ $\bar{f}\circ \tau=\bar f$, and $\|\frac{1}{n}\sum_{k=0}^{n-1}T^kf-\bar{f}\|_p\to 0$ as $n\to \infty$.

For $p=2$ this is just von Neumann's mean ergodic theorem. Using this One can easily prove the result for $1\leq p<2$. But how to prove the statement for $2<p<\infty$? Also is it true for $p=\infty$?