Metric outer measure

In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space such that
for every pair of positively separated subsets A and B of X.

Construction of metric outer measures

Let τ : Σ → 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.
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.

For any sequence of subsets A_n, n ∈ N, of X with
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.