In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that
μ ( A ∪ B ) = μ ( A ) + μ ( B ) {\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
μ ( E ) = lim δ → 0 μ δ ( E ) , {\displaystyle \mu (E)=\lim _{\delta \to 0}\mu _{\delta }(E),}
where
μ δ ( E ) = inf { ∑ i = 1 ∞ τ ( C i ) | C i ∈ Σ , diam ( C i ) ≤ δ , ⋃ i = 1 ∞ C i ⊇ E } , {\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\},}
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
τ ( C ) = diam ( C ) s , {\displaystyle \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 An, n ∈ N, of X with
A 1 ⊆ A 2 ⊆ ⋯ ⊆ A = ⋃ n = 1 ∞ A n , {\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
μ ( A ) = sup n ∈ N μ ( A n ) . {\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,
μ ( A ∪ B ) = μ ( A ) + μ ( B ) . {\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.
References Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.
