Lebesgue outer measure
\PMlinkescapephrase
extension theorem
Let $S$ be a subset of $\mathbb{R}$, let $L(I)$ be the traditional definition of the length of an interval $I\subseteq \mathbb{R}$: If $I=(a,b)$, then $L(I)=ba$. Finally, let $M$ be the set consisting of the values
$$\sum _{A\in C}L(A)$$ 
for all possible countable^{} collections^{} of open intervals $C$ that covers $S$ (that is, $S\subseteq \cup C$). Then the Lebesgue outer measure of $S$ is defined by:
$${m}^{*}(S)=inf(M)$$ 
Note that $(\mathbb{R},\mathcal{P}(\mathbb{R}),{m}^{*})$ is an outer measure^{} space (http://planetmath.org/OuterMeasure2). In particular:

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Lebesgue outer measure is defined for any subset of $\mathbb{R}$ (and $\mathcal{P}(\mathbb{R})$ is a $\sigma $algebra).

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${m}^{*}(A)\ge 0$ for any $A\subseteq \mathbb{R}$, and ${m}^{*}(\mathrm{\varnothing})=0$.

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If $A$ and $B$ are disjoint sets, then ${m}^{*}(A\cup B)\le {m}^{*}(A)+{m}^{*}(B)$. More generally, if $\u27e8{A}_{i}\u27e9$ is a countable sequence of disjoint sets, then ${m}^{*}\left(\bigcup {A}_{i}\right)\le \sum {m}^{*}({A}_{i})$. This property is known as countable subadditivity and is weaker than countable additivity^{}. In fact, ${m}^{*}$ is not countably additive.
Lebesgue outer measure has other nice properties:

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The outer measure of an interval is its length: ${m}^{*}((a,b))=ba$.

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${m}^{*}$ is translationinvariant. That is, if we define $A+y$ to be the set $\{x+y:x\in A\}$, we have ${m}^{*}(A)={m}^{*}(A+y)$ for any $y\in \mathbb{R}$.
The outer measure satisfies all the axioms of a measure except (countable) additivity. However, it is countably additive when one restricts to at least the Borel sets, as this is the usual construction of Borel measure. This result is roughly contained in the Caratheodory Extension theorem.
Title  Lebesgue outer measure 
Canonical name  LebesgueOuterMeasure 
Date of creation  20130322 11:48:15 
Last modified on  20130322 11:48:15 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  14 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 28A12 
Synonym  outer measure 
Related topic  Infimum^{} 
Related topic  LebesgueMeasure 
Related topic  ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength 
Related topic  CaratheodorysLemma 
Related topic  ConstructionOfOuterMeasures 