von Neumann ergodic theorem for $L_p$

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 $$2? Also is it true for $$p=\infty$$?

• Something's odd about the properties of $\overline{f}$: you probably mean $\overline{f} \circ \tau = \overline{f}$ instead of $\overline{f} \circ \tau = f$, right? – Jochen Glueck May 30 '20 at 20:06
• @Jochen. Yes. Edited. – A beginner mathmatician May 30 '20 at 20:11

• From Krengel only I am reading this. There is no proof of this fact. He only stated this. As I mentioned I could prove for $1\leq p<2$ but not for $p>2$. Also now I see that he has given only a hint for $p=\infty$. But after lot of try I could not solve that too. – A beginner mathmatician May 30 '20 at 18:59
• @Abeginnermathmatician: It is indeed in Krengel's book: For $p \in (1,\infty)$ it is an immediate consequence of Theorem 2.1.2 on page 73. Right after the theorem, Krengel explains why it is also true for $p = 1$ (as a consequence of Theorem 2.1.1 on page 72). – Jochen Glueck May 30 '20 at 20:14
• @Abeginnermathmatician: For a $p=\infty$ counterexample, consider the shift on 2 symbols with the Bernoulli (1/2,1/2) measure and $f(x)=x_0$. Then the pointwise limit of the averages is 1/2 almost everywhere; for each $n$, there is a set of positive measure (namely the set of $x$’s with $x_0=...=x_{n-1}=0$) where the difference between the average and the limit is 1/2. – Anthony Quas May 30 '20 at 20:15