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*.

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*_{δ}(*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.

Let *μ* be a metric outer measure on a metric space (*X*, *d*).

- For any sequence of subsets
*A*_{n},*n*∈**N**, of*X*with

- and such that
*A*_{n}and*A*\*A*_{n+1}are positively separated, it follows that

- All the
*d*-closed subsets*E*of*X*are*μ*-measurable in the sense that they satisfy the following version of Carathéodory's criterion: for all sets*A*and*B*with*A*⊆*E*and*B*⊆*X*\*E*,

- Consequently, all the Borel subsets of
*X*— those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are*μ*-measurable.