3

This is Petersen 2.2.1. Let $\{T(t,x)\}_{t \in \mathbb{R}}$ be a family of one-parameter invertible measure preserving transformations on $(X, \mathcal{M}, \mu)$ a measure space. Let $f : X \rightarrow \mathbb{R}$ be an $L^1(\mu)$ function. The goal is to formulate and prove the Maximal Ergodic Theorem for flows. My idea is that this looks like the Hardy-Littlewood theorem, so try to just shove it into that. That is, let $$A_r(f)(x) = \frac{1}{|B(r,0)|} \int_{B(r,0)} f(T(s,x)) d\lambda(s),$$ and let $$f^*(x) = \sup_{r > 0} A_r(f)(x).$$ Then the formulation would be $$ \int_{\{f^* > 0\}} f d\mu \geq 0,$$ and the proof would be something like the Hardy-Littlewood theorem (except now you're going to have to use the invertible measure preserving transformation to move between $\mathbb{R}$ and $X$ so that you can use the Vitali covering lemma). Is this a reasonable formulation/idea?

It also asks for the Pointwise Ergodic theorem, but if my hunch is correct for the above then I'm pretty sure this is going to be something like the Lebesgue differentiation theorem.

Based on the discussion in Ergodic Theorem and flow I have a feeling this might not be the correct approach, but I am still curious.

User203940
  • 2,473
  • Maybe to add a little to the discussion... The Wikipedia article on Maxima Functions somewhat reaffirms that I at least have said something that makes sense, but it still doesn't tell me whether this strategy of just stealing the proof of Hardy-Littlewood is valid. https://en.wikipedia.org/wiki/Maximal_function#Maximal_functions_in_ergodic_theory – User203940 Sep 25 '20 at 13:01

1 Answers1

0

I don't think the Hardy-Littlewood theorem is used in the standard setting to prove that $\int_{f^* > 0} fd\mu \ge 0$. See Walter's book on ergodic theory. All that's used is that (the analogue of) $A_r$ is a positive operator.

mathworker21
  • 34,399
  • I figured this was the case. They seemed similar enough that I was hopeful it was just a consequence of Hardy-Littlewood. Thanks for the input! – User203940 Sep 30 '20 at 13:14