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

[math]\displaystyle{ \mu (A \cup B) = \mu (A) + \mu (B) }[/math]

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

[math]\displaystyle{ \mu (E) = \lim_{\delta \to 0} \mu_{\delta} (E), }[/math]

where

[math]\displaystyle{ \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\}, }[/math]

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

[math]\displaystyle{ \tau(C) = \operatorname{diam} (C)^s,\, }[/math]

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

[math]\displaystyle{ A_{1} \subseteq A_{2} \subseteq \dots \subseteq A = \bigcup_{n = 1}^{\infty} A_{n}, }[/math]

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

[math]\displaystyle{ \mu (A) = \sup_{n \in \mathbb{N}} \mu (A_{n}). }[/math]

• 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,

[math]\displaystyle{ \mu (A \cup B) = \mu (A) + \mu (B). }[/math]

• 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

• Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.