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    Properties of Lebesgue outer measure

    PR-LF7SW

    Lebesgue outer measure is monotone, countably subadditive, outer regular, and additive on sets at positive distance.Suggest a slogan

    Let \(m_*\) be Lebesgue outer measure on \(\RR^n\) [@SS05].

    1. If \(E\subseteq F\subseteq\RR^n\), then \(m_*(E) \leq m_*(F)\).

    2. If \(E_i\subseteq\RR^n\) for \(i\geq1\), then \(m_*\big(\bigcup_{i\geq1} E_{i}\big) \leq \sum_{i\geq1} m_*(E_{i})\).

    3. For every \(E\subseteq\RR^n\) and \(\varepsilon>0\) there exists an open set \(G\supseteq E\) with \(m_*(G) \leq m_*(E) + \varepsilon\).

    4. If \(A,B\subseteq\RR^n\) satisfy \(\dist(A,B)\coloneqq\inf\theset{\abs{a-b} : a\in A,\ b\in B}>0\), then \(m_*(A\cup B) = m_*(A) + m_*(B)\).

    In (d), positive distance is sufficient but not necessary: for \(A\coloneqq[0,1)\) and \(B\coloneqq[1,2]\) in \(\RR\), \(\dist(A,B)=0\) and \(m_*(A\cup B)=2=m_*(A)+m_*(B)\). For disjoint sets in general, additivity fails: there are disjoint \(A,B\subseteq\RR\) with \(m_*(A\cup B)<m_*(A)+m_*(B)\), constructed from a non-measurable Vitali set.

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