= High-density interval for outer measure
{title2=$m^*(E\cap I)>\lambda|I|,\quad0<\lambda<1$}
= Outer-measure high-density interval
{synonym}
For any $E\subseteq\mathbb R$ with positive <Lebesgue outer measure> and any $0<\lambda<1$, some bounded nonempty open <interval> $I$ satisfies the displayed inequality. The set $E$ need not be a <Lebesgue measurable set>.
To prove this, first choose a bounded portion $F=E\cap[-N,N]$ with $0<a=m^*(F)<\infty$, using <countable subadditivity>. Choose a countable open-interval cover $(I_j)$ of $F$ with $\sum_j|I_j|<a/\lambda$. If $m^*(E\cap I_j)\leq\lambda|I_j|$ for every $j$, then
$$
a\leq\sum_jm^*(F\cap I_j)\leq\lambda\sum_j|I_j|<a,
$$
a contradiction. For measurable $E$ this gives the <high-density interval in a positive-measure subset of the real line> without requiring the full <Lebesgue density theorem>.
Back to article page
