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; 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 and packing measures are obtained.

Properties of metric outer measures

Let μ be a metric outer measure on a metric space.