A standard proposition in measure theory is that if $\displaystyle E$ is any Lebesgue measurable set in $\displaystyle \mathbb{R}$ with $\displaystyle \lambda(E)>0$, then for any $\displaystyle \epsilon>0$ there is a finite, nontrivial interval $\displaystyle J=[a,b]$ such that $\displaystyle \lambda(E\cap J)>(1-\epsilon)\lambda(J)$.

To generalize this, suppose $\displaystyle E\subseteq\mathbb{R}^n$ with $\displaystyle \lambda(E)>0$. Then why for any $\displaystyle \epsilon>0$ is there some box $\displaystyle J=(a_1,a_1+\delta]\times\cdots\times(a_n,a_n+\delta]$ such that $\displaystyle \lambda(E\cap J)>(1-\epsilon)\lambda(J)$? Thank you.