Timeline for von Neumann ergodic theorem for $L_p$

7 events

when toggle format	what		by	license	comment
May 31, 2020 at 6:12	vote	accept	A beginner mathmatician
May 31, 2020 at 4:56	history	edited	Yuval Peres	CC BY-SA 4.0	added 453 characters in body
May 30, 2020 at 20:15	comment	added	Anthony Quas		@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.
May 30, 2020 at 20:14	comment	added	Jochen Glueck		@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).
May 30, 2020 at 18:59	comment	added	A beginner mathmatician		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.
May 30, 2020 at 18:37	review	Low quality posts
May 30, 2020 at 18:37
May 30, 2020 at 18:21	history	answered	Yuval Peres	CC BY-SA 4.0