I cannot solve a seemingly simple question about the Lebesgue measure. Suppose we have a measurable set of positive measure $E$ and suppose there exists an open interval $I_E$ such that $m( \cap I_E)≥\frac{9}{10}|I_E|$.
Let $0<\delta<|I_E|$ , will there always exist an open interval $J$ of length $\delta$ such that $J \subset I_E$ and $m(E \cap I_E \cap J) \ge \frac{9}{10}\delta$ ? I think such a $J$ must surely exist, however, I cannot prove it.
You cannot prove it because it is not true. Let $E=[0,1]\backslash[\frac{9}{20},\frac{11}{20}]$, let $I_E=(0,1)$, and let $\delta=\frac{8}{9}$. Then there is no interval $J$ inside of $(0,1)$ of length $\frac{8}{9}$ with this property, since every such interval must completely contain $[\frac{9}{20},\frac{11}{20}]$, and therefore satisfies $$m(E\cap I_E\cap J)=\frac{8}{9}-\frac{1}{10}=\frac{71}{90}<\frac{8}{10}=\frac{9}{10}\cdot \frac{8}{9}.$$