Metric outer measure explained

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) = \lim _{\delta} \mu _{\delta} (E), where\mu _{\delta} (E) = inf \left\{ \left. \sum ^{inf ty} _{i =1} \tau (C _{i} ) \right| C _{i} \in \Sigma, \operatorname{diam} (C _{i} ) \leq \delta, c up ^{inf ty} _{i =1} 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 μ_δ(E) increases as δ decreases.)

For the function τ one can use

\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 A_n, n ∈ N, of X with

A _{1} \subseteq A _{2} \subseteq ... \subseteq A = c up ^{inf ty} _{n =1} A _{n} ,

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

\mu (A) = \sup _{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

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