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Suppose I have a sequence of real measurable functions $f_n$ converging to $f$ on $[0,1]$ pointwise. For any $\epsilon$ greater than zero, can we choose $n$ large enough such that the length of the set of points satisfying $|f_n-f|\ge\epsilon$ can be made arbitrarily small?

More precisely, given $S=\{x\in[0,1]: |f_n(x)-f(x)|\ge \epsilon\}$, can we choose $N$ large enough such that for all $\delta>0$ and $n\ge N$, the measure of $S$ is less than $\delta$?

I came up with this while solving an analysis problem I thought it was really interesting but I couldn't get anything in motion. I feel like it should be true, but I think this question + Baire's theorem means game over. I'm far out my league here to say so decisively though, so I have come here for an opinion. Please let me know if I can clarify the question further.

smorsgabord
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    Why do you claim that $S$ can be written as a disjoint union of open intervals along with isolated points? – José Carlos Santos Aug 14 '21 at 18:55
  • @JoséCarlosSantos Sorry, isolated points was the wrong terminology. If you can see what I'm getting at and can think of a better way of phrasing it than my edit, please feel free to edit it yourself – smorsgabord Aug 14 '21 at 18:56
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    The assertion that $S$ can be written that way definitely fails when the $f_n$ and $f$ are not continuous. It also fails even under a continuity assumption (in general it can be a pretty arbitrary closed set, which certainly doesn't have the structure asserted), although there might be a modification that works in that case. – Greg Martin Aug 14 '21 at 19:16
  • Probably a sensible choice is to assume that the $f_n$ and $f$ are measurable, so that $S$ itself is measurable, and to talk about the measure of $S$ rather than the length. – Greg Martin Aug 14 '21 at 19:19
  • @GregMartin What about now? I think thats solid now. – smorsgabord Aug 14 '21 at 19:23
  • Oh I didn't see your last comment. I was not aware that $f_n$ and $f$ measurable implies $S$ is(was) measurable. Although, I don't know any measure theory, so if the new definition still doesn't work, I'll revert it and we'll roll witih your idea. – smorsgabord Aug 14 '21 at 19:24
  • Measure theory is the standard way to make your intuition about "percentage of points" precise, and you seem to be on your way to reinventing it on your own with your attempts at defining $L(S)$. There are various standard pitfalls to avoid, though, and it would probably be a more efficient use of your time (though perhaps not as fun) to consult a text in measure theory -- the giants are right there, just climb onto their shoulders ... – Troposphere Aug 14 '21 at 19:38
  • @Troposphere You are right, I have been ignorant, how could I expect help when using my own drivel instead of standard definitions. – smorsgabord Aug 14 '21 at 19:45
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    @smorsgabord: My apologies if my comment came across as putting you down; that was not my intention. On the contrary, it shows laudable initiative to attempt to come up with a workable definition by yourself -- I just thought you might find it useful to know that the very thing you're grasping for does already exist under such-and-such name, in case you're in a hurry. In particular, reading a standard text would enable you to skip past a number of false starts and almost-but-not-quite workable ideas that occupied the ancients for decades. – Troposphere Aug 14 '21 at 19:54
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    One of the key lessons from measure theory is that one cannot construct a concept of "length" of an arbitrary set which satisfies all of the "natural" nice properties one would want it to have, unless one accepts that there can be "weird" sets it won't work for. That realization took quite a long time to come to terms with for the first mathematicians to encounter it, and it makes for faster progress to avail oneself of their hard-won insight -- both in terms of what is not possible, the simplest proofs of why it's not possible, and what the known options for what to do instead are. – Troposphere Aug 14 '21 at 19:58
  • @Troposphere Thank you for taking the time to explain that to me. – smorsgabord Aug 14 '21 at 20:09
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    What you are essentially asking for is: "does pointwise convergence imply convergence in measure?" This is indeed true. See Here: https://en.m.wikipedia.org/wiki/Convergence_in_measure – PhoemueX Aug 14 '21 at 20:37
  • @PhoemueX Could you narrow down that link please, "pointwise" is not mentioned on that page. Is there a proof of what you say? – smorsgabord Aug 16 '21 at 19:06
  • See e.g. here for several proofs: https://math.stackexchange.com/questions/1506763/does-a-point-wise-convergence-of-measurable-functions-imply-a-convergence-in-fi – PhoemueX Aug 16 '21 at 20:32

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