---
tag: BJ314H17HC
kind: concept
name: Outer measure
msc: [28A12]
sources:
  - url: https://en.wikipedia.org/wiki/Outer_measure
same_as:
  wikidata: Q258374
  wikipedia: Outer measure
---

## Summary

An outer measure assigns a size in $[0, \infty]$ to every subset of a set, not only to the sets of a $\sigma$-algebra.
It is monotone and countably subadditive, but in general not additive.
Through [[Carathéodory's criterion|JDGQ135YK2]] every outer measure produces a $\sigma$-algebra on which it is a complete [[measure|PR11BZQQPG]].
This is the standard way to build Lebesgue measure and Hausdorff measures.

## Explanations

### L1 human

To measure an odd-shaped set, cover it by simple pieces whose size is known, and take the smallest total size of such a cover.
This over-estimate is the outer measure.
It can be applied to any set at all, but for very wild sets it is not additive.
The union of two disjoint sets can have an outer measure smaller than the sum of their outer measures.
For example, Lebesgue outer measure covers a subset of $\R$ by countably many intervals and takes the infimum of the total length.
Every countable set gets outer measure $0$, because the $n$-th point can be covered by an interval of length $\varepsilon / 2^n$.