Lebesgue density theorem

For every Lebesgue measurable set $A\subseteq {\mathbb{R}}^n$, its Lebesgue density is $1$ at almost every point of $A$ and $0$ at almost every point of its complement.

If a measurable set $E\subseteq \mathbb{R}$ has positive measure, then for every $\epsilon >0$ there is a bounded interval $I$ such thatChoose a density-one point of a finite positive-measure portion of $E$ using the Lebesgue density theorem, and take a sufficiently small interval centred there.