Definition:Lebesgue Measure/Real Number Line

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Definition

For a given set $S \subseteq \R$, let $\set {I_n}$ be a countable set of open intervals such that:

$S \subseteq \bigcup \set {I_n}$

For the power set $\powerset \R$ of the real numbers $\R$, construct a function $\mu^*: \powerset \R \to \R_{>0}$ as:

$\ds \map {\mu^*} S = \inf \set {\sum_{n \mathop \in \N} \map l {I_n} : \set {I_n} : S \subseteq \bigcup_{n \mathop \in \N} I_n}$

where:

the infimum ranges over all such sets $\set {I_n}$

$\map l {I_n}$ is the length of the interval $I_n$.

Then $\mu^*$ is known as the Lebesgue outer measure and can be shown to be an outer measure.

Also known as

When the domain of a Lebesgue outer measure $\mu^*$ is restricted to the set $\mathfrak M$ of Lebesgue-measurable sets, $\mu^*$ is instead written as $\mu$ and is simply referred to as the Lebesgue measure.

Also see

• Construction of Outer Measure where a Lebesgue outer measure is shown to be an outer measure

• Results about Lebesgue measures can be found here.

Source of Name

This entry was named for Henri Léon Lebesgue.

Sources

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