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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e473499451f41cd1e883247b1efedc0469b2d65b)
for every pair of positively separated subsets A and B of X.
Construction of metric outer measures
[edit]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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e233f77a6aa476e399e410cdde31fda6babefae2)
where
![{\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\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f04238fa39c001948c3c6400555b605cc3169fc1)
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)=\operatorname {diam} (C)^{s},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/570c39e5b69f6ac2db7d13522e3139f3c2f563af)
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
[edit]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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e59b49a45b6b262c94a2713db41cb64dbdedf1a)
- 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d941c07f372d30f0ce18f9fec71d81b6b062d57)
- 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b07b7af3dcd6b3b16e5a91548c631ca8904e32f8)
- Consequently, all the Borel subsets of X — those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are μ-measurable.
- Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.