---
tag: Q8EXG7EKYW
kind: statement
role: theorem
name: Carathéodory's extension theorem
msc: [28A12]
uses: [PYH82FMXQ2, PV0XXSWFC7, PR11BZQQPG, 0SV8K0TDGY, WEBBT18D4X]
about: [PV0XXSWFC7, PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem
same_as:
  wikidata: Q2631152
  wikipedia: Carathéodory's extension theorem
---

## Statement

Let $X$ be a set and let $\mathcal{S}$ be a [[semiring of sets|PYH82FMXQ2]] on $X$.
Let $\mu_0 \colon \mathcal{S} \to [0, \infty]$ be a [[premeasure|PV0XXSWFC7]], that is, $\mu_0(\varnothing) = 0$ and $\mu_0\big(\bigcup_n A_n\big) = \sum_n \mu_0(A_n)$ for every sequence of pairwise disjoint sets $A_n \in \mathcal{S}$ whose union lies in $\mathcal{S}$.
Let $\sigma(\mathcal{S})$ be the [[sigma-algebra|0SV8K0TDGY]] generated by $\mathcal{S}$.

1. There is a [[measure|PR11BZQQPG]] $\mu$ on $\sigma(\mathcal{S})$ with $\mu(A) = \mu_0(A)$ for all $A \in \mathcal{S}$.
2. If moreover $X = \bigcup_{n} X_n$ for a sequence of sets $X_n \in \mathcal{S}$ with $\mu_0(X_n) < \infty$, then this measure $\mu$ is unique.

## Remarks

For existence, define $\mu^*(B) = \inf \set{ \sum_n \mu_0(A_n) : A_n \in \mathcal{S}, \ B \subseteq \bigcup_n A_n }$.
This is an outer measure, and every set of $\mathcal{S}$ satisfies the Carathéodory splitting condition.
Countable additivity of $\mu_0$ gives $\mu^* = \mu_0$ on $\mathcal{S}$.
So $\mu^*$ restricted to the Carathéodory-measurable sets is the required extension.
Uniqueness follows from Dynkin's $\pi$-$\lambda$ theorem, since $\mathcal{S}$ is closed under finite intersections.
Example: the half-open intervals $(a, b]$ form a semiring on $\R$, and $\mu_0((a, b]) = b - a$ is a premeasure; the theorem gives Lebesgue measure on the Borel sets, and it is unique.
Uniqueness can fail without the finiteness hypothesis in clause 2.
On $\Q$, take the semiring of sets $(a, b] \cap \Q$ and the premeasure that is $\infty$ on nonempty sets and $0$ on $\varnothing$.
It extends both to counting measure and to $2$ times counting measure on the power set of $\Q$.

## Explanations

### L1 human

It is often easy to say how big simple sets are, such as intervals or boxes.
The theorem says that if this size is countably additive on the simple sets, it can be extended to a measure on every set built from them by countable operations.
If the whole space is covered by countably many simple sets of finite size, there is only one way to do it.

### L2 human

This is the main tool to construct measures: Lebesgue measure, Lebesgue–Stieltjes measures, product measures and measures on sequence spaces all come from it.
The only work left in each case is to check that the given set function is countably additive on the semiring.
A version for an algebra of sets, with the extension given by an explicit outer measure, is the [[Hahn–Kolmogorov theorem|FN2WRRF6V9]].
The proof goes through an [[outer measure|BJ314H17HC]] and [[Carathéodory's criterion|JDGQ135YK2]].