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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The concept of a metric outer measure is a way to extend the notion of "size" or "measure" of subsets of a metric space. It builds on the idea of open covers and the associated infimum of sums of the measures of covering sets. Here’s how it works in a structured manner: ### Definition Let \((X, d)\) be a metric space.