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 $p=\infty$?



von Neumann ergodic theorem for $L_p$
False for p infinite. True for finite. See e.g. the book by Krengel, Ergodic theorems. Other sources (that also go further) are [1, Sec. I.2.1] or [2, Theorem 8.8]. 


[1] T. Eisner, Stability of operators and operator semigroups, Operator Theory:
Advances and Applications, vol. 209, Birkh¨auser Verlag, Basel, 2010.


[2] T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator theoretic aspects
of ergodic theory, Graduate Texts in Mathematics, vol. 272, Springer, Cham,
2015. <http://www.math.uni-leipzig.de/~eisner/book-EFHN.pdf>


