Timeline for Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 23, 2021 at 23:19 | vote | accept | Julian Newman | ||
| Oct 22, 2021 at 22:47 | answer | added | Yuval Peres | timeline score: 10 | |
| Oct 22, 2021 at 5:56 | answer | added | Anthony Quas | timeline score: 5 | |
| Oct 22, 2021 at 2:11 | history | edited | Julian Newman | CC BY-SA 4.0 |
added clarity to proof of BET
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| Oct 22, 2021 at 1:35 | history | edited | Julian Newman | CC BY-SA 4.0 |
added clarity to proof of BET
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| Oct 22, 2021 at 1:21 | comment | added | Julian Newman | I've now added the short proof of Birkhoff's ergodic theorem for bounded observables. | |
| Oct 22, 2021 at 1:20 | history | edited | Julian Newman | CC BY-SA 4.0 |
Added "in the appropriate manner"
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| Oct 22, 2021 at 0:29 | comment | added | Julian Newman | @KConrad Yes - I can take a closer look, but even that proof seemed significantly more involved than the proof I currently have for the case of bounded observables. | |
| Oct 21, 2021 at 23:41 | comment | added | KConrad | Have you looked at the paper of Katznelson and Weiss "A Simple Proof of Some Ergodic Theoems" (Israel J. Math.) 42 (1982), 291-296? Roughly speaking, it involves replacing an $L^1$-function $f$ with $\min(f,M)$ for a positive constant $M$, which amounts to replacing $f$ with a certain bounded $L^1$-function. | |
| Oct 21, 2021 at 23:20 | comment | added | Julian Newman | @AnthonyQuas Even in the former case, although that technically wouldn't fulfil what I've asked, I'm still very interested, as the very short proof that I have in mind for bounded observables already basically consists of proving a suitable version of the maximal ergodic theorem (from which the result then follows immediately by making a simple choice of function to which to apply that maximal ergodic theorem) | |
| Oct 21, 2021 at 23:12 | history | edited | Julian Newman | CC BY-SA 4.0 |
added condition
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| Oct 21, 2021 at 23:09 | comment | added | Julian Newman | @AnthonyQuas Is this maximal ergodic theorem specifically for $P_n$ being the $n$-th Birkhoff average under a measure-preserving transformation, or is it for any sequence of Markov operators fulfilling the conditions I described? | |
| Oct 21, 2021 at 23:01 | comment | added | Anthony Quas | I'm guessing you don't consider the use of the maximal ergodic theorem to be elementary. One formulation of the maximal ergodic theorem is that the subset of $L^1$ consisting of those functions for which $P_nf$ converges pointwise is closed. | |
| Oct 21, 2021 at 21:38 | history | asked | Julian Newman | CC BY-SA 4.0 |