PlanetMath: Lebesgue outer measure

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Lebesgue outer measure

(Definition)

Let be a subset of $\mathbb{R}$ , let be the traditional definition of the length of an interval $I \subseteq \mathbb{R}$ : If , then . Finally, let be the set consisting of the values

$\displaystyle \sum_{A\in C}L(A)$

for all possible countable collections of open intervals that covers (that is, $S \subseteq \cup C$ ).

Then the Lebesgue outer measure of is defined by:

$\displaystyle m^{*}(S) = \inf(M)$

Note that $(\mathbb{R},\mathcal{P}(\mathbb{R}),m^{*})$ is an outer measure space. 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 and 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 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.

"Lebesgue outer measure" is owned by yark. [ full author list (4) | owner history (3) ]

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Other names:

outer measure

Keywords:

real analysis

Pronunciation (guide):

Lebesgue: /l*-beg''''''''''''''''''''''''''''''''/

Attachments:: proof that the outer (Lebesgue) measure of an interval is its length (Proof) by Simone

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Cross-references: Borel measure, Borel sets, additivity, measure, countably additive, countable additivity, subadditivity, sequence, disjoint, covers, open intervals, collections, countable, interval, subset
There is 1 reference to this entry.

This is version 9 of Lebesgue outer measure, born on 2001-10-18, modified 2006-09-11.
Object id is 341, canonical name is LebesgueOuterMeasure.
Accessed 14830 times total.

Classification:

AMS MSC:

28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

Pending Errata and Addenda

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