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**ramdayal9** ($\displaystyle m$ is the exterior/outer measure). Let $\displaystyle E \subset \mathbb{R}$ with $\displaystyle m(E)>0$. Prove that for each $\displaystyle 0 < \alpha < 1$, there exists an open interval $\displaystyle I$ such that $\displaystyle m(E \cap I) \geq \alpha m(I)$.

Im not sure how to proceed....Ive tried starting with the definition of the outer measure and considering a cover of open intervals but im not making any progress. Ive also looked at contradiction, but not sure if im getting anywhere with that. Any help will be appreciated

Thanks