Lebesgue outer measure

A closed cube in \(\RR^n\) is a product \(Q=\prod_{i=1}^n[a_i,a_i+\ell]\) with \(\ell\geq 0\), and its volume is \(\abs{Q}\coloneqq\ell^n\). The outer measure of a set \(E\subseteq\RR^n\) is \[ m_*(E) \coloneqq \inf \theset{\sum_{i=1}^\infty \abs{Q_i} \suchthat Q_1, Q_2, \ldots \text{ closed cubes with } E\subseteq\bigcup_{i=1}^\infty Q_i} \in [0,\infty]. \]