Metric outer measure

Remarks

Example: Lebesgue outer measure $\lambda^*$ on $\R^n$ is a metric outer measure. Let $d(A, B) = \delta > 0$. A cover of $A \cup B$ by boxes can be refined into boxes of diameter less than $\delta$, and then no box meets both $A$ and $B$. So the cover splits into a cover of $A$ and a cover of $B$, which gives $\ge$; subadditivity gives $\le$. Non-example: on $\R$, put $\mu^*(\varnothing) = 0$ and $\mu^*(A) = 1$ for every nonempty $A$. This is an outer measure, but $\mu^*(\set{0, 1}) = 1 \ne \mu^*(\set{0}) + \mu^*(\set{1})$ although $d(\set{0}, \set{1}) = 1 > 0$. Carathéodory's theorem on metric outer measures: an outer measure on a metric space is a metric outer measure if and only if every Borel set is Carathéodory measurable for it.