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Let $E$ be a set of positive Lebesgue measure on the real line. Let $1>\epsilon>0$ be given. Show that there exists an interval $I$ such that $m(E\cap I)>\epsilon m(I)$ where $m$ is the Lebesgue measure on the real line.

I tried answering by contradiction method without success. Then tried writing the inequality as $m(I\backslash E)<(1-\epsilon)m(I)$ which also didn't help much either.

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If $E$ is a set of positive Lebesgue measure, then there exists a Borel set $D$ of type $F_{\delta}$ such that $\mu(E\Delta D)=0$. So it suffice to prove this for $F_{\sigma}$ type sets. And it is clear that we can reduce this to a counterable intersection of open sets such that one contains the other.

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    You can find at Real and Complex Analysis, walter rudim, pg 1412012-12-28