In mathematics, a metric outer measure is an

outer measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...

''μ'' defined on the subsets of a given

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

(''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 → , +∞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_ \mu_ (E),

where :

\mu_ (E) = \inf \left\,

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 (C)^s,\,

where ''s'' is a positive constant; this ''τ'' is defined on the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

of all subsets of ''X''. By

Carathéodory's extension theorem In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ- ...

, the outer measure can be promoted to a full measure; the associated measure ''μ'' is the ''s''-dimensional

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...

. More generally, one could use any so-called

dimension function In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" ...

. This construction is very important in

fractal geometry In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

, since this is how the

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_ \subseteq A_ \subseteq \dots \subseteq A = \bigcup_^ A_,

:and such that ''A''_''n'' and ''A'' \ ''A''_''n''+1 are positively separated, it follows that ::

\mu (A) = \sup_ \mu (A_).

* 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

* {{Measure theory Measures (measure theory) Metric geometry