Metric outer measure
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
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
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.
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
- and such that An and A \ An+1 are positively separated, it follows that
- 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,
- Consequently, all the Borel subsets of X — those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are μ-measurable.