Lebesgue measurable sets

Let \(m_*\) be the outer measure on \(\RR^n\). A subset \(E\subseteq \RR^n\) is Lebesgue measurable if for every \(\varepsilon> 0\) there exists an open set \(O \supseteq E\) such that \(m_*(O\setminus E) < \varepsilon\). For a Lebesgue measurable set \(E\), its Lebesgue measure is \(m(E) \coloneqq m_*(E)\).