Lebesgue inner measure ($m_{\ast }(E)=\sup \{m(K):K\subseteq E,K\text{ compact}\}$)

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 $E$,Indeed, every compact $K\subseteq E$ gives $m^{\ast }(I\setminus E)\le m(I)-m(K)$. 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 $m^{\ast }(E)<\infty$, 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 $m_{\ast }(E)=m^{\ast }(E)=\infty$ alone is insufficient: adjoining a bounded nonmeasurable subset to a disjoint half-line makes both quantities infinite without making the union measurable.