In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that

\( {\displaystyle \mu (A\cup B)=\mu (A)+\mu (B)} \)

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

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

where

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

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 μδ(E) increases as δ decreases.)

For the function τ one can use

\( {\displaystyle \tau (C)=\mathrm {diam} (C)^{s},\,} \)

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 An, n ∈ N, of X with

\( {\displaystyle A_{1}\subseteq A_{2}\subseteq \dots \subseteq A=\bigcup _{n=1}^{\infty }A_{n},} \)

and such that An and A \ An+1 are positively separated, it follows that

\( {\displaystyle \mu (A)=\sup _{n\in \mathbb {N} }\mu (A_{n}).} \)

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,

\( {\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).} \)

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

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