Neha Patil (Editor)

Metric outer measure

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In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that

Contents

μ ( A ∪ B ) = μ ( A ) + μ ( 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 ) ,

where

μ δ ( E ) = inf { ∑ i = 1 ∞ τ ( C i ) | C i ∈ Σ , d i a m ( C i ) ≤ δ , ⋃ i = 1 ∞ C i ⊇ E } ,

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 ) = d i a m ( C ) s ,

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.

This construction is very important in fractal geometry, since this is how the Hausdorff and packing measures are obtained.

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.

    • References

    Metric outer measure Wikipedia
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