# Lebesgue outer measure

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 $L(I)=b-a$. 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 $(\mathbb{R},\mathcal{P}(\mathbb{R}),m^{*})$ is an outer measure space (http://planetmath.org/OuterMeasure2). In particular:

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

• $m^{*}(A)\geq 0$ for any $A\subseteq\mathbb{R}$, and $m^{*}(\emptyset)=0$.

• If $A$ and $B$ are disjoint sets, then $m^{*}(A\cup B)\leq m^{*}(A)+m^{*}(B)$. More generally, if $\langle A_{i}\rangle$ is a countable sequence of disjoint sets, then $m^{*}\left(\bigcup A_{i}\right)\leq\sum m^{*}(A_{i})$. 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 $y\in\mathbb{R}$.

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 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