Proof of $L^{\infty}$ mean ergodic theorem

Question

One can use Birkhoff's ergodic theorem to, not only prove, but extend Mean ergodic theorem to $L^{p}$ functions where $p\in\left[1,\infty\right).$

This questions shows this while ask about a proof of Peter Walters' book:

Questions to the proof of the $L^p$ Ergodic Theorem of Von Neumann

On the other hand, for the case $p=\infty$ there are counter-examples, such as:

https://mathoverflow.net/questions/303697/a-counterexample-for-the-mean-ergodic-theorem-in-l-infty

and

Counterexample for the $L^p$ Ergodic Theorem of Von Neumann when $p=\infty$

So my question is: where does the proof shown in Walters' book (the first link) fails for $p=\infty?$

Thanks in advance!

Answer 1

The proof uses Lebesgue's theorem for the convergence of integrals (in this case $L^p$-norms). This does not hold for the convergence in $L^\infty$: Consider, e.g., the unit interval with Lebesgue measure and the indicator functions $f_n=I_{(0,1/n)}$ which converge to $0$ pointwise and are bounded by the constant function $1$, but nevertheless satisfy $\|f_n-0\|_\infty=1$.