Lebesgue outer measure
= Lebesgue outer measure
{title2=$m^*$}
The resulting <Lebesgue measure> is the restriction of this outer measure to the sets satisfying the measurability criterion.
Lebesgue outer measure on $\mathbb R$ is
$$
m^*(E)=\inf\left\{\sum_{n=1}^{\infty}|I_n|:E\subseteq\bigcup_{n=1}^{\infty}I_n\right\},
$$
where the infimum is over countable interval covers. The <Caratheodory measurability criterion> selects the Lebesgue-measurable sets.