Lebesgue outer measure

extension theorem

Let $S$ be a subset of $ℝ$, let $L(I)$ be the traditional definition of the length of an interval $I\subseteq ℝ$: If $I=(a,b)$, then $L(I)=b-a$. 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 $(ℝ,𝒫(ℝ),m^{*})$ is an outer measure space (http://planetmath.org/OuterMeasure2). In particular:

• Lebesgue outer measure is defined for any subset of $ℝ$ (and $𝒫(ℝ)$ is a $\sigma$-algebra).

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

• If $A$ and $B$ are disjoint sets, then $m^{*}(A\cup B)\le m^{*}(A)+m^{*}(B)$. More generally, if $⟨A_i⟩$ 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:

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

• $m^{*}$ is translation-invariant. 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 $y\in ℝ$.

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	2013-03-22 11:48:15
Last modified on	2013-03-22 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