Metric outer measure
In mathematics, a metric outer measure is an outer measure "μ" defined on the subsets of a given metric space ("X", "d") such that for every pair of positively separated subsets "A" and "B" of "X". Construction of metric outer measures. Let "τ" : Σ → [0, +∞] be a set function defined on a class Σ of subsets of "X" containing the empty set ∅, such that "τ"(∅) = 0. One can show that the set function "μ" defined by where is not only an outer measure, but in fact a metric outer measure as well . (Some authors prefer to take a supremum over "δ" > 0 rather than a limit as "δ" → 0; the two give the same result, since "μ"("E") increases as "δ" decreases.) For the function "τ" one can use where "s" is a positive constant; this "τ" is defined on the power set of all subsets of "X". By Carathéodory's extension theorem, the outer measure can be promoted to a full measure; the associated measure "μ" is the "s"-dimensional Hausdorff measure. More generally, one could use any so-called dimension function. This construction is very important in fractal geometry, since  this is how the Hausdorff measure is obtained. The packing measure is superficially similar, but is obtained in a different manner, by packing balls inside a set, rather than covering the set. Properties of metric outer measures. Let "μ" be a metric outer measure on a metric space ("X", "d").