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    $\begingroup$ Thanks! I was about to delete my question in light of <a href="mathoverflow.net/questions/8972/…> (which has good information on related stuff) and then I saw your very quick answer. $\endgroup$
    – Alexander Pruss
    Commented Mar 25, 2013 at 14:48
  • $\begingroup$ Link was bad: mathoverflow.net/questions/8972/… $\endgroup$
    – Alexander Pruss
    Commented Mar 25, 2013 at 14:48
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    $\begingroup$ Alternatively, if $X\subset[0,1]$ has positive measure, find $x\in[0,1]$ such that $[0,x]\cap X$ and $[x,1]\cap X$ both have positive measure. Then find $x_0$ and $x_1$ such that $0<x_0<x<x_1<1$ and $X$ has positive measure in each of the intervals $[0,x_0]$, $[x_0,x]$, $[x,x_1]$, $[x_1,1]$. Continue subdividing, and find continuum-many points in $X$. $\endgroup$
    – Sean Eberhard
    Commented Mar 25, 2013 at 14:49
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    $\begingroup$ Aha, yes, that question includes information about the non-measurable case, which I ignored. $\endgroup$
    – Sean Eberhard
    Commented Mar 25, 2013 at 14:51
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    $\begingroup$ @Sean: Concerning your comment about repeated subdivision, did you intend the "continuum-many points" in the last sentence to be the limits of the subdivision points? If so, how do you know they're in $X$. If not, which continuum-many points did you mean? $\endgroup$
    – Andreas Blass
    Commented Mar 25, 2013 at 17:24