# Definition:Lebesgue Measure

## Definition

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

Let $\mathcal B \left({\R^n}\right)$ be the Borel $\sigma$-algebra on $\R^n$.

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

Any measure $\mu$ extending $\lambda^n$ to $\mathcal B \left({\R^n}\right)$ is called **$n$-dimensional Lebesgue measure**.

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

- $\mu \restriction_{\mathcal J_{ho}^n} = \lambda^n$

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 $\left\{{I_n}\right\}$ be a countable set of open intervals such that

- $S \subseteq \bigcup I_n$

For the power set $\mathcal P \left({\R}\right)$ of the real numbers $\R$, construct a function $\mu^*: \mathcal P \left({\R}\right) \to \R_{>0}$ as:

- $\displaystyle \mu^* \left({S}\right) = \inf \left\{{\sum_{n \mathop \in \N} l \left({I_n}\right) : \left\{{I_n}\right\} : S \subseteq \bigcup_{n \mathop \in \N} I_n}\right\}$

where the infimum ranges over all such sets $\left\{{I_n}\right\}$, and $l \left({I_n}\right)$ 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 $\left({\R, \mathfrak M, \mu}\right)$ 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): $4.8$