Timeline for Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2021 at 17:02 | comment | added | Bill Johnson | @GeraldEdgar: WLOG the functions are simple functions. | |
| Aug 6, 2021 at 16:11 | comment | added | Gerald Edgar | "Rearrange to be decreasing" ... That is not $f \circ T$ as in the OP. For example, $f(x) = 4x(1-x)$ in $[0,1]$ where each value in $(0,1)$ occurs twice. How do we "rearrange" it to be decreasing? We cannot use measure-preserving bijection for that. | |
| Aug 6, 2021 at 15:43 | comment | added | Nate River | @fedja Oh, I think that is true - in that case we just rearrange all the $f_i$ to be decreasing and we are done then. | |
| Aug 6, 2021 at 12:55 | comment | added | fedja | Isn't any set of decreasing uniformly bounded functions pre-compact in $L^1$? | |
| Aug 6, 2021 at 11:52 | history | edited | Gerald Edgar | CC BY-SA 4.0 |
misprint
|
| Aug 6, 2021 at 9:19 | history | edited | Nate River | CC BY-SA 4.0 |
added 64 characters in body
|
| Aug 6, 2021 at 8:45 | history | edited | Nate River | CC BY-SA 4.0 |
added 69 characters in body
|
| Aug 6, 2021 at 8:44 | comment | added | Nate River | Ah, sorry my bad. I will modify it, and also write the definition in the original post. | |
| Aug 6, 2021 at 8:27 | comment | added | Jochen Glueck | Thanks for the clarification! This is actually called weak (rather than weak-$*$) convergence. | |
| Aug 6, 2021 at 8:22 | comment | added | Nate River | Sorry, I mean that $\int f_i g \ d\mu \to \int fg \ d\mu$ for all $g \in L^\infty$. | |
| Aug 6, 2021 at 6:53 | comment | added | Jochen Glueck | What do you mean by weak-$*$ convergence in $L^1$? | |
| Aug 6, 2021 at 2:17 | history | asked | Nate River | CC BY-SA 4.0 |