= Metric outer measure
{title2=$d(A,B)>0\Longrightarrow\mu^*(A\cup B)=\mu^*(A)+\mu^*(B)$}
An <outer measure> on a <metric space> is metric when positively separated sets satisfy the displayed equality. Every <Borel set> is then measurable under <Caratheodory measurability criterion>. <Lebesgue outer measure> is metric: refine every box of a covering into finitely many boxes of diameter smaller than the separation. Each refined box meets at most one of the two sets, splitting the cover without changing its total volume. Taking infima proves the additive lower bound; outer subadditivity gives equality.
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