= Lebesgue inner measure
{c}
{title2=$m_*(E)=\sup\{m(K):K\subseteq E,\ K\text{ compact}\}$}
The <Lebesgue inner measure> approximates a subset of the real line from within by <compact sets>, while <Lebesgue outer measure> approximates it from outside by countable interval covers. For a finite closed interval $I$ containing $E$,
$$
m_*(E)=m(I)-m^*(I\setminus E).
$$
Indeed, every compact $K\subseteq E$ gives $m^*(I\setminus E)\leq m(I)-m(K)$. Conversely, an open cover of $I\setminus E$ with total length arbitrarily close to its <outer measure> has compact complement in $I$ lying inside $E$, giving the reverse inequality in the limit.
For $m^*(E)<\infty$, equality of inner and outer measures holds precisely when $E$ is <Lebesgue measurable>. For an unbounded set, measurability is equivalently checked on all bounded intersections $E\cap[-n,n]$. Equality $m_*(E)=m^*(E)=\infty$ alone is insufficient: adjoining a bounded nonmeasurable subset to a disjoint half-line makes both quantities infinite without making the union measurable.
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