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Is this question acceptable?: Does pointwise convergeness imply uniform convergence on a large subset?

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    Is it OK to ask the following on MO?


    Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.

    1. Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$?

    2. Is there a subset $A$ of $[0,1]$ of cardinality the continuum so that $f_n$ converges uniformly on $A$?

    Background: Egoroff's theorem implies that the answer to (2) is yes if all $f_n$ are Lebesgue measurable. It is not hard to show that the answer to (1) is yes if you change "uncountable" to "infinite".

    Motivation: I thought about this question while teaching real analysis this term but could not solve it even after looking at some books, googling, and asking some colleagues who are much smarter than I, so I assigned it as a problem (well, an extra credit problem) to my class. Unfortunately, no one gave me a solution.

    Apology: OK, this is not really a research level question, but it also seems too advanced for other possible boards, and I imagine I can get a reference here from someone.
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    @Bill: your question is certainly acceptable on MO.

    I don't have time at the moment to write out a careful explanation of why, but roughly: if a post-PhD mathematician has a question about a certain subject (within or without their core areas of research expertise) and has made at least some effort to answer it in more conventional ways -- e.g. through a literature search, asking colleagues -- then it is appropriate to ask this question on MO. Indeed, this situation is perhaps the main reason for MO's existence.

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    I do not see why it would be a problem to have this question.
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    +1 Pete.

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    +1 Pete for the explanation.

    Looks like a great question to me. You should cut out the apology. My favorite questions on MO are usually exactly this kind of curiosity.

    • CommentAuthordeane.yang
    • CommentTimeNov 11th 2010
     
    Pete's absolutely right, especially since Bill is somewhat more than just a "post-PhD mathematician".
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    I agree -- this is a fantastic question.
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    I could have waited and posted this as a comment on the question, but I did some Googling and came up with something that looks relevant, Theorem 10 quoted and linked to below from Morgan's *Point set theory*. It cites Sierpiński, but I can't tell what works are cited because the preview won't let me see that page in the references.

    "The existence of a linear set having the power of the continuum that is concentrated on a denumerable set is equivalent to the existence of a pointwise convergent sequence of functions of a real variable that does not converge uniformly on any uncountable set."

    http://books.google.com/books?id=WwmvxtDlz9UC&lpg=PA124&ots=lcdSy9gacd&dq=point%20set%20theorem%20morgan&pg=PA88#v=onepage&q&f=false
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    Thanks, Jonas. With that reference I will be able to find out what is known.
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    Still, please post this as a question; not many people who visit the site visit meta and they will end up missing it otherwise.
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    Still, please post this as a question; not many people who visit the site visit meta and they will end up missing it otherwise.

    agreed

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    OK.

    Jonas, will you please post your answer on MO?
    • CommentAuthorWill Jagy
    • CommentTimeNov 11th 2010
     
    I would not ordinarily view meta. But I saw your question. It moved me.
    • CommentAuthorWill Jagy
    • CommentTimeNov 11th 2010
     
    convergeness ??
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    Bill, I did so after I saw you asked me to. Unfortunately I have nothing further to add.

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