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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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    Are you sure that what you're trying to prove is true? What about $E$ being the union of two disjoint intervals and $\epsilon$ close to one?2012-12-28
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    See [this](http://math.stackexchange.com/questions/103306/a-pair-of-lebesgue-measure-questions-concerning-sets-of-strictly-positive-measur/103331) post.2012-12-28
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    This is better than I solution.2012-12-28
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    I've removed [tag:qualifying-exam] tag from the questions where it was used; [1](http://math.stackexchange.com/questions/266460/construct-an-entire-function-given-its-zero-set), [2](http://math.stackexchange.com/questions/266568/lebesgue-measure-related-question), [3](http://math.stackexchange.com/questions/266857/difficulty-understanding-branch-of-the-logarithm). Meta-tags and dependent tags are generally [discouraged](http://blog.stackoverflow.com/2010/08/the-death-of-meta-tags/).2012-12-30
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    If you think the tag would be useful, feel free to ask about opinion of other users at meta or to join the discussion in [tagging chatroom](http://chat.stackexchange.com/transcript/message/7449932#7449932).2012-12-30

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