---
tag: WD1AJ2X325
kind: definition
name: Outer measure
msc: [28A12]
defines: BJ314H17HC
uses: [00E0YZCERH, 0GSZ7B0CGR]
sources:
  - url: https://en.wikipedia.org/wiki/Outer_measure
---

## Definition

Let $X$ be a set and let $\mathcal{P}(X)$ be its [[power set|00E0YZCERH]].
An *outer measure* on $X$ is a map $\mu^* \colon \mathcal{P}(X) \to [0, \infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:

1. $\mu^*(\varnothing) = 0$.
2. If $A \subseteq B \subseteq X$, then $\mu^*(A) \le \mu^*(B)$.
3. For every sequence $A_1, A_2, \ldots$ of subsets of $X$,
$$\mu^*\Big(\bigcup_{n=1}^{\infty} A_n\Big) \le \sum_{n=1}^{\infty} \mu^*(A_n).$$

## Remarks

Example: Lebesgue outer measure on $\R$ is
$$\lambda^*(A) = \inf \Big\{ \sum_{n} (b_n - a_n) : A \subseteq \bigcup_{n} (a_n, b_n) \Big\},$$
the infimum over all countable covers of $A$ by open intervals.
Clause 1 holds since $\varnothing$ is covered by intervals of arbitrarily small total length.
Clause 2 holds since a cover of $B$ also covers $A$.
Clause 3 follows by covering each $A_n$ within $\varepsilon / 2^n$ of $\lambda^*(A_n)$.
Non-example: on $X = \set{1, 2}$, the map with $\mu(\varnothing) = 0$, $\mu(\set{1}) = \mu(\set{2}) = 1$ and $\mu(X) = 3$ is monotone, but it fails clause 3 because $3 > 1 + 1$.
Some authors, such as Federer, call outer measures simply measures.
The outer measures defined here are the ones to which [[Carathéodory's criterion|JDGQ135YK2]] applies.