 Solution

The Lebesgue outer measure isThe Lebesgue inner measure is $m_{\ast }(E)=\sup \{m(K):K\subseteq E,K\text{ compact}\}$, with $m_{\ast }(\varnothing )=0$. For a bounded set inside a finite interval $I$, this is equivalently $|I|-m^{\ast }(I\setminus E)$.

A set is Lebesgue measurable if it satisfies Carathéodory's criterion:For finite outer measure this is equivalent to equality of inner and outer measure. For a general unbounded set, an equivalent local criterion is $m_{\ast }(E\cap [-n,n])=m^{\ast }(E\cap [-n,n])$ for every positive integer $n$. Merely writing $m_{\ast }(E)=m^{\ast }(E)=\infty$ would not establish measurability.