Timeline for near ergodic theory question

9 events

when toggle format	what		by	license	comment
May 26, 2023 at 7:14	comment	added	Augusto Santos		@OlegZubelewicz: It is Ok.
May 25, 2023 at 19:12	vote	accept	Oleg Zubelewicz
May 25, 2023 at 19:12	comment	added	Oleg Zubelewicz		Thanks, I asked a stupid question
May 25, 2023 at 19:00	comment	added	Augusto Santos		@OlegZubelewicz: I am not sure I understand your latter comment, but I updated the answer to render this step clearer.
May 25, 2023 at 19:00	history	edited	Augusto Santos	CC BY-SA 4.0	added 290 characters in body
May 25, 2023 at 18:43	comment	added	Oleg Zubelewicz		$A_kf:=\frac{1}{k}\sum_{j=0}^{k-1}fT^j;\quad A_k(fT)=A_kf-f/k+\frac{1}{k}fT^k$
May 25, 2023 at 18:29	comment	added	Augusto Santos		@OlegZubelewicz: You are missing that $g_{0}= \limsup_{k\rightarrow \infty} \frac{1}{k} \sum_{i=0}^{k-1} fT^{i}= \limsup_{k\rightarrow \infty} \frac{1}{k} \sum_{i=0}^{\ell-1} fT^{i}+\limsup_{k\rightarrow \infty} \frac{1}{k} \sum_{i=\ell}^{k-1} fT^{i}$. Since the first term is zero (regardless of whether $f\in L^{\infty}$ or not, since it is a finite sum of finite elements) and the second is $g_{\ell}$, you have that $g_0=g_{\ell}$, for any $\ell \in \mathbb{N}$.
May 25, 2023 at 18:16	comment	added	Oleg Zubelewicz		Yes, and it remains to show that $g_1=g_0$. I understand that just for $f\in L^\infty$
May 25, 2023 at 17:35	history	answered	Augusto Santos	CC BY-SA 4.0