A problem about Lebesgue measurable set I'm doing exercise in "REAL ANALYSIS" of Folland and got stuck on this problem. I got no clue on how to find the set $I$. Hope someone can help me solve this. Thanks so much


Suppose $m$ is Lebesgue measure and $L$ is its domain. If $E \in L$ and $m(E) \gt 0$, for any $\alpha < 1$, prove that there is an open interval $I$ such that $m(E \bigcap I) \gt \alpha m(I)$



Hint: Use regularity of Lebesgue measure to find an open set with the desired property. Then use the fact that any open set is a disjoint union of intervals.
