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| In [[mathematics]], in particular in [[measure theory]], an '''outer measure''' or ''exterior measure'' is a [[function (mathematics)|function]] defined on all subsets of a given [[Set (mathematics)|set]] with values in the [[extended real line|extended real numbers]] satisfying some additional technical conditions. A general theory of outer measures was first introduced<ref>See pp379, C.D. Aliprantis, K.C. Border, ''Infinite Dimensional Analysis'', 3rd ed, Springer 2006. ISBN 3-540-29586-0</ref> by [[Constantin Carathéodory]]<ref>[[Constantin Carathéodory|C. Carathéodory]], ''Vorlesungen über reelle Funktionen'', 1st ed, Berlin: Leipzig 1918, 2nd ed, New York: Chelsea 1948.</ref> to provide a basis for the theory of [[measurable set]]s and [[sigma additive|countably additive]] measures. Carathéodory's work on outer measures found many applications in measure-theoretic [[set theory]] (outer measures are for example used in the proof of the fundamental [[Carathéodory's extension theorem]]), and was used in an essential way by [[Felix Hausdorff|Hausdorff]] to define a dimension-like metric [[invariant (mathematics)|invariant]] now called [[Hausdorff dimension]].
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| Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in '''R''' or balls in '''R'''<sup>3</sup>. One might expect to define a generalized measuring function φ on '''R''' that fulfils the following requirements:
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| # Any interval of reals [''a'', ''b''] has measure ''b'' − ''a''
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| # The measuring function φ is a non-negative extended real-valued function defined for all subsets of '''R'''.
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| # Translation invariance: For any set ''A'' and any real ''x'', the sets ''A'' and ''A+x'' have the same measure (where <math>A+x = \{ a+x: a\in A\}</math>)
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| # [[Countable additivity]]: for any [[sequence]] (''A''<sub>''j''</sub>) of pairwise [[disjoint set|disjoint subsets]] of '''R'''
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| ::<math> \varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i).</math>
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| It turns out that these requirements are incompatible conditions; see [[non-measurable set]]. The purpose of constructing an ''outer'' measure on all subsets of ''X'' is to pick out a class of subsets (to be called ''measurable'') in such a way as to satisfy the countable additivity property.
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| == Formal definitions ==
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| An outer measure on a set {{math|X}} is a function
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| :<math>\varphi: 2^X \rightarrow [0, \infty], </math>
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| defined on all subsets of {{math|X}} ({{math|2<sup>X</sup>}} is another notation for the power set), that satisfies the following conditions:
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| *Null empty set: The [[empty set]] has zero outer measure (''see also: [[measure zero]]'').
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| :: <math> \varphi(\varnothing) = 0</math>
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| * [[monotonic function|Monotonicity]]: For any two subsets {{math|A}} and {{math|B}} of {{math|X}},
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| :: <math>A\subseteq B\quad\text{implies}\quad\varphi(A) \leq \varphi(B).</math>
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| * Countable [[subadditivity]]: For any sequence {{math|{A<sub>j</sub>} }} of subsets of {{math|X}} (pairwise disjoint or not),
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| :: <math> \varphi\left(\bigcup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \varphi(A_j).</math>
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| This allows us to define the concept of measurability as follows: a subset {{math|E}} of {{math|X}} is φ-measurable (or [[Carathéodory]]-measurable by φ) [[iff]] for every subset {{math|A}} of {{math|X}}
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| :<math> \varphi(A) = \varphi(A \cap E) + \varphi(A \cap E^c). </math>
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| '''Theorem'''. The φ-measurable sets form a [[sigma algebra|σ-algebra]] and φ restricted to the measurable sets is a countably additive [[complete measure]]. For a proof of this theorem see the Halmos reference, section 11. This method is known as the Carathéodory construction and is one way of arriving at the concept of [[Lebesgue measure]] that is so important for measure theory and the theory of [[integral]]s.
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| == Outer measure and topology ==
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| Suppose {{math|(X, d)}} is a [[metric space]] and {{math|φ}} an outer measure on {{math|X}}. If {{math|φ}} has the property that
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| :<math> \varphi(E \cup F) = \varphi(E) + \varphi(F)</math>
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| whenever
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| :<math> d(E,F) = \inf\{d(x,y): x \in E, y \in F\} > 0, </math>
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| then {{math|φ}} is called a [[metric outer measure]].
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| '''Theorem'''. If {{math|φ}} is a metric outer measure on {{math|X}}, then every Borel subset of {{math|X}} is {{math|φ}}-measurable. (The [[Borel algebra|Borel set]]s of {{math|X}} are the elements of the smallest {{math|σ}}-algebra generated by the open sets.)
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| == Construction of outer measures ==
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| There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as '''Method I''' and '''Method II'''.
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| === Method I===
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| Let {{math|X}} be a set, {{math|C}} a family of subsets of {{math|X}} which contains the empty set and {{math|p}} a non-negative extended real valued function on {{math|C}} which vanishes on the empty set.
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| '''Theorem'''. Suppose the family {{math|C}} and the function {{math|p}} are as above and define
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| :<math> \varphi(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C\biggr\}.</math>
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| That is, the [[infimum]] extends over all sequences {{math|{A<sub>i</i>} }} of elements of {{math|C}} which cover {{math|E}}, with the convention that the infimum is infinite if no such sequence exists. Then {{math|φ}} is an outer measure on {{math|X}}.
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| ===Method II ===
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| The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose {{math|(X, d)}} is a metric space. As above {{math|C}} is a family of subsets of {{math|X}} which contains the empty set and {{math|p}} a non-negative extended real valued function on {{math|C}} which vanishes on the empty set. For each {{math|δ > 0}}, let
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| :<math>C_\delta= \{A \in C: \operatorname{diam}(A) \leq \delta\} </math>
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| and | |
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| :<math> \varphi_\delta(E) = \inf \biggl\{ \sum_{i=0}^\infty p(A_i)\,\bigg|\,E\subseteq\bigcup_{i=0}^\infty A_i,\forall i\in\mathbb N , A_i\in C_\delta\biggr\}.</math>
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| Obviously, {{math|φ<sub>δ</sub> ≥ φ<sub>δ'</sub>}} when {{math|δ ≤ δ'}} since the infimum is taken over a smaller class as {{math|δ}} decreases. Thus
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| :<math> \lim_{\delta \rightarrow 0} \varphi_\delta(E) = \varphi_0(E) \in [0, \infty]</math>
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| exists (possibly infinite).
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| '''Theorem'''. {{math|φ<sub>0</sub>}} is a metric outer measure on {{math|X}}.
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| This is the construction used in the definition of [[Hausdorff measure]]s for a metric space.
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| ==See also==
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| *[[Inner measure]]
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| == References ==
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| <references/>
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| * [[Paul Halmos|P. Halmos]], ''Measure theory'', D. van Nostrand and Co., 1950
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| * M. E. Munroe, ''Introduction to Measure and Integration'', Addison Wesley, 1953
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| * A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, ''Introductory Real Analysis'', Dover Publications, New York, 1970 ISBN 0-486-61226-0
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Outer_measure Outer measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Caratheodory_measure Caratheodory measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| [[Category:Measures (measure theory)]]
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