Timeline for Does pointwise convergence imply uniform convergence on a large subset?
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2021 at 14:18 | comment | added | Kr Dpk | @BillJohnson This is very late considering the time since you asked this question. I have nothing to add to the answers or the question. But it'll be of great knowledge to me if you could elaborate on a comment that you made in your "background". You stated that in part (1) if we simply have the condition of $A$ being infinite, then (1) can be "easily" proved. I am embarassed to state that i couldnt do that. It would mean a lot if you shed some light on the stated comment. Does it hold for any sequence of pointwise convergent functions ? No need for complete proof, just some direction suffice | |
| Feb 27, 2011 at 3:12 | answer | added | Alberto Santini | timeline score: 5 | |
| Nov 12, 2010 at 23:03 | comment | added | George Lowther | The following generalization also sounds interesting. If $f_n$ is a sequence of real valued functions on a set A converging pointwise to zero, for what cardinalities $\kappa$ can we guarantee the existence of a subset $B\subseteq A$ with $\vert B\vert=\kappa$ on which it converges uniformly? And for what $\kappa$ can we guarantee the existence of such a subset with $\vert A\setminus B\vert=\kappa$? | |
| Nov 12, 2010 at 22:15 | history | edited | Bill Johnson | CC BY-SA 2.5 |
added 107 characters in body
|
| Nov 12, 2010 at 22:14 | vote | accept | Bill Johnson | ||
| Nov 12, 2010 at 14:36 | answer | added | Eric Wofsey | timeline score: 28 | |
| Nov 12, 2010 at 14:30 | comment | added | Bill Johnson | @ SJR: By replacing $f_n$ with $g_n := \sup_{k \ge n} f_k$, the questions reduce to the case where the sequence is pointwise decreasing to zero, so a result for a subsequence gives the same result for the sequence. Thanks for the reference. | |
| Nov 12, 2010 at 5:15 | comment | added | Sidney Raffer | The paper "Small Combinatorial Cardinal Characteristics and Theorems of Ergorov and Blumberg" by Krzysztof Ciesielski and Janusz Pawlikowski, Real Analysis Exchange, p905-912, might be useful here. It seems to imply that the answer to your question is "yes" for some SUBSEQUENCE of the $f_n$, subject to the consistency with ZFC of certain hypotheses about Ramsey ultrafilters and dominating numbers. The paper is on Project Euclid. Here's a link: projecteuclid.org/… | |
| Nov 12, 2010 at 5:11 | answer | added | Nate Eldredge | timeline score: 12 | |
| Nov 12, 2010 at 5:08 | answer | added | Andrey Rekalo | timeline score: 16 | |
| Nov 12, 2010 at 4:50 | history | edited | Will Jagy | CC BY-SA 2.5 |
what does convergeness mean
|
| Nov 12, 2010 at 4:40 | comment | added | Will Jagy | My one remaining desire, an answer from Jonas Meyer, has been fulfilled. Now I rest. | |
| Nov 12, 2010 at 4:27 | answer | added | Jonas Meyer | timeline score: 22 | |
| Nov 12, 2010 at 4:25 | comment | added | Will Jagy | convergeness ?? | |
| Nov 12, 2010 at 4:04 | comment | added | Yemon Choi | Of course, for your questions it is only the cardinality of [0,1] that is important, not any topological or measure-space structure we might usually put on it. I assume you phrased it this way to highlight the contrast with Egoroff's theorem? | |
| Nov 12, 2010 at 4:02 | comment | added | Will Jagy | This is the best question I have ever seen. | |
| Nov 12, 2010 at 4:01 | history | asked | Bill Johnson | CC BY-SA 2.5 |