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 $\JJ_{ho}^n$.
A 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.
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.
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): measure
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.8$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): measure