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Does there exist a set $E\subset [0,1]$ with $m(E)<1$ such that $m(E\cap I)\geq m(I)/2$ for all measurable sets $I\subset [0,1]$? I am not able to construct one, but it seems possible. Any help would be appreciated! Thanks!

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Is $I$ an interval or an arbitrary measurable subset of $[0,1]$? – JSchlather Dec 10 '12 at 3:48
Duplicate of A Lebesgue measure question - the question is different, but the answers address this question (after taking complements). – Nate Eldredge Dec 10 '12 at 3:57
@Jacob. $I$ is an arbitrary measurable subset of $[0,1]$. – stephen Dec 10 '12 at 5:40
@Nate. I am not sure I see how the answers address this question. What do you mean by taking complements? – stephen Dec 10 '12 at 5:43
@Nate: I too fail to see how the linked question contains an answer for the present one. – Martin Argerami Dec 10 '12 at 6:04
up vote 1 down vote accepted

Such set cannot exist.

Since $m(E)<1$, $m(E^c)>0$. Then $m(E\cap E^c)=0<m(E^c)/2$.

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