Lebesgue outer measure

For a closed cube \(Q=\prod_{i=1}^n[a_i,a_i+\ell]\subseteq\RR^n\), write \(\abs{Q}\coloneqq\ell^n\) for its volume. The outer measure of \(E\subseteq\RR^n\) is \[ m_*(E) \coloneqq \inf \theset{ \sum_{i=1}^\infty \abs{Q_i} \suchthat Q_1,Q_2,\ldots \text{ closed cubes with } E\subseteq\bigcup_{i=1}^\infty Q_i}. \]