---
tag: 2Z2QVWZ262
kind: object
name: Lebesgue measure
msc: [28A12, 28A75]
instance_of: [PR11BZQQPG]
uses: [BJ314H17HC, 0XPR1E85W0, 0WQXY32Q8D]
sources:
  - url: https://en.wikipedia.org/wiki/Lebesgue_measure
same_as:
  wikidata: Q827230
  wikipedia: Lebesgue measure
---

## Description

Let $n \ge 1$.
A box in $\R^n$ is a product $R = [a_1, b_1] \times \cdots \times [a_n, b_n]$ with $a_i \le b_i$, and its volume is $\operatorname{vol}(R) = (b_1 - a_1) \cdots (b_n - a_n)$.
Lebesgue [[outer measure|BJ314H17HC]] on [[Euclidean space|0XPR1E85W0]] $\R^n$ is
$$\lambda^*(A) = \inf \Big\{ \sum_{k=1}^{\infty} \operatorname{vol}(R_k) : R_1, R_2, \ldots \text{ boxes}, \ A \subseteq \bigcup_{k} R_k \Big\} \quad \text{for } A \subseteq \R^n.$$
A set $E \subseteq \R^n$ is Lebesgue measurable if $\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \setminus E)$ for all $A \subseteq \R^n$.
Lebesgue measure $\lambda$ (or $\lambda^n$) is $\lambda^*$ restricted to the Lebesgue measurable sets.

## Remarks

By [[Carathéodory's criterion|JDGQ135YK2]] the Lebesgue measurable sets form a $\sigma$-algebra and $\lambda$ is a complete [[measure|PR11BZQQPG]] on it.
Every [[Borel set|0WQXY32Q8D]] is Lebesgue measurable, and $\lambda(R) = \operatorname{vol}(R)$ for every box $R$; so $\lambda([a, b]) = b - a$ on $\R$.
Lebesgue measure is invariant under translations and rotations, and $\lambda(tA) = \abs{t}^n \lambda(A)$ for $t \in \R$.
Up to a constant factor, it is the only translation-invariant measure on the Borel sets of $\R^n$ that is finite on bounded sets.
Every countable set has measure $0$, and so does the Cantor set, which is uncountable.
Lebesgue measurable sets are regular: for every measurable $E$ and $\varepsilon > 0$ there are a closed set $F$ and an open set $U$ with $F \subseteq E \subseteq U$ and $\lambda(U \setminus F) < \varepsilon$.
Not every subset of $\R^n$ is Lebesgue measurable; a Vitali set is an example.

## Explanations

### L1 human

Lebesgue measure is length on the line, area in the plane and volume in space, extended to as many sets as possible.
To measure a set, cover it with countably many boxes and take the smallest possible total volume.
The interval $[0, 3]$ gets length $3$, a disk of radius $r$ gets area $\pi r^2$, and the set of rational numbers gets length $0$.