= Solution
{c}
The <Lebesgue outer measure> is
$$
m^*(E)=\inf\left\{\sum_j|I_j|:E\subseteq\bigcup_j I_j,\ I_j\text{ open intervals}\right\}.
$$
The <Lebesgue inner measure> is $m_*(E)=\sup\{m(K):K\subseteq E,\ K\text{ compact}\}$, with $m_*(\varnothing)=0$. For a bounded set inside a finite interval $I$, this is equivalently $|I|-m^*(I\setminus E)$.
A set is <Lebesgue measurable> if it satisfies <Carathéodory's criterion>:
$$
\boxed{m^*(A)=m^*(A\cap E)+m^*(A\setminus E)\quad\text{for every }A\subseteq\mathbb R.}
$$
For finite outer measure this is equivalent to equality of inner and outer measure. For a general unbounded set, an equivalent local criterion is $m_*(E\cap[-n,n])=m^*(E\cap[-n,n])$ for every positive integer $n$. Merely writing $m_*(E)=m^*(E)=\infty$ would not establish measurability.
Back to article page
