Metric outer measure
# 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

    $\mu (A\cup B)=\mu (A)+\mu (B)$

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

    $\mu (E)=\lim _{\delta \to 0}\mu _{\delta }(E),$

where

    $\mu _{\delta }(E)=\inf \left\{\left.\sum _{i=1}^{\infty }\tau (C_{i})\right|C_{i}\in \Sigma ,\operatorname {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\},$

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

    $\tau (C)=\operatorname {diam} (C)^{s},\,$

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).

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

       $A_{1}\subseteq A_{2}\subseteq \dots \subseteq A=\bigcup _{n=1}^{\infty }A_{n},$
    and such that $A_n$ and A \ $A_{n+1}$ are positively separated, it follows that
       $\mu (A)=\sup _{n\in \mathbb {N} }\mu (A_{n}).$
* 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,

       $\mu (A\cup B)=\mu (A)+\mu (B).$
* Consequently, all the Borel subsets of X — those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are μ-measurable.
