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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

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

where

${\mu }_{\delta }(E)=\inf \{\sum _{i=1}^{\mathrm{\infty }}\tau (C_i)|C_i\in \mathrm{\Sigma },\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(C_i)\le \delta ,\bigcup _{i=1}^{\mathrm{\infty }}{C}_i\supseteq E\},$

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

$\tau (C)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(C{)}^s,$

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 (X, d).

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

$A_1\subseteq A_2\subseteq \dots \subseteq A=\bigcup _{n=1}^{\mathrm{\infty }}{A}_n,$

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

$\mu (A)={\sup }_{n\in \mathbb{\mathbb{N}}}\mu (A_n).$

• 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.

References

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