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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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    @MartinArgerami: The "general fact" in xen's answer, applied to $E^c$ instead of $E$, and with, say, $c=3/4$, shows that there is an interval $I$ with $m(E^c \cap I) \ge \frac{3}{4} m(I)$, which means $m(E \cap I) \le \frac{1}{4} m(I)$. Of course if $I$ is allowed to be any measurable set, this becomes trivial as in Martin Argerami's answer below.2012-12-10

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Such set cannot exist.

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