Basic properties of Lebesgue outer measure

Let \(m_*\) be the outer measure on \(\RR^n\).

1. Monotonicity. If \(E_1\subseteq E_2\subseteq\RR^n\), then \(m_*(E_1)\leq m_*(E_2)\).

2. Countable subadditivity. If \(E=\bigcup_{j=1}^\infty E_j\) with \(E_j\subseteq\RR^n\), then \(m_*(E)\leq\sum_{j=1}^\infty m_*(E_j)\).

3. Approximation from above by open sets. For every \(E\subseteq\RR^n\), \(m_*(E)=\inf\theset{m_*(O)\suchthat O\supseteq E \text{ open}}\).

4. Additivity for almost disjoint cubes. If \(E=\bigcup_{j=1}^\infty Q_j\) is a countable union of closed cubes that are pairwise almost disjoint, then \(m_*(E)=\sum_{j=1}^\infty\abs{Q_j}\).