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I'm working on a problem from Stein and Shakarchi's Real Analysis, and it asks to construct a measurable subset $E\subset [0,1]$ such that for any non-empty sub-interval $I$ in $[0,1]$, both $E\cap I$ and $E^c \cap I$ have positive measure. The problem gives the hint "to consider a Cantor-like set of positive measure, and add in each of the intervals that are omitted in the first step of its construction, another Cantor-like set. Continue this procedure indefinitely."

I don't quite understand the hint. Am I just taking the collection of open subsets of $[0,1]$ removed from the Cantor-like set? How could these be another Cantor-like set, since they're open?

Nate
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  • Let $I_1,I_2,\dots$ be the intervals omitted in constructing the first Cantor-like set. By "add in each of the intervals . . . another Cantor-like set" what they mean is, construct a Cantor-like subset of $I_1$, a Cantor-like subset of $I_2$, and so on. – bof Feb 18 '19 at 00:08
  • For another solution of that problem, see the answer to this question: https://math.stackexchange.com/questions/641466/creating-a-lebesgue-measurable-set-with-peculiar-property – bof Feb 18 '19 at 00:11
  • The intervals omitted are open, so do we take the closure of each open set to construct a Cantor-like set? And then union those closures to get our desired set? – Nate Feb 18 '19 at 01:07
  • Ah, I misunderstood. We're making a Cantor-like subset of those removed intervals. Right? – Nate Feb 18 '19 at 01:09

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