Definition:Lebesgue Measure

Definition

Let $\JJ_{ho}^n$ be the set of half-open $n$-rectangles.

Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$.

Let $\lambda^n$ be the $n$-dimensional Lebesgue pre-measure on $\JJ_{ho}^n$.

Any measure $\mu$ extending $\lambda^n$ to $\map \BB {\R^n}$ is called $n$-dimensional Lebesgue measure.

That is, $\mu$ is an $n$-dimensional Lebesgue measure if and only if it satisfies:

$\mu \restriction_{\JJ_{ho}^n} = \lambda^n$

where $\restriction$ denotes restriction.

Usually, this measure is also denoted by $\lambda^n$, even though this may be considered abuse of notation.

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Lebesgue Measure on the Reals

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.

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

Also see

• Existence and Uniqueness of Lebesgue Measure justifying the fact that one may speak simply about (the) $n$-dimensional Lebesgue measure.

• Measure Space from Outer Measure, where it is shown that $\struct {\R, \mathfrak M, \mu}$ is a measure space.

• Results about Lebesgue measures can be found here.

Source of Name

This entry was named for Henri Léon Lebesgue.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.8$