High-density interval in a positive-measure subset of the real line
= High-density interval in a positive-measure subset of the real line
If a measurable set $E\subseteq\mathbb R$ has positive measure, then for every $\varepsilon>0$ there is a bounded interval $I$ such that
$$
m(E\cap I)>(1-\varepsilon)m(I).
$$
Choose 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.