---
tag: FN2WRRF6V9
kind: statement
role: theorem
name: Hahn–Kolmogorov theorem
msc: [28A12]
uses: [PFMGZ7988V, PV0XXSWFC7, BJ314H17HC, PR11BZQQPG, 0SV8K0TDGY, WEBBT18D4X]
about: [PV0XXSWFC7, BJ314H17HC]
sources:
  - url: https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem
---

## Statement

Let $X$ be a set, let $\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on $X$, and let $\mu_0 \colon \mathcal{A} \to [0, \infty]$ be a [[premeasure|PV0XXSWFC7]].
For $B \subseteq X$ put
$$\mu^*(B) = \inf \Big\{ \sum_{n=1}^{\infty} \mu_0(A_n) : A_n \in \mathcal{A}, \ B \subseteq \bigcup_{n=1}^{\infty} A_n \Big\}.$$
Then:

1. $\mu^*$ is an [[outer measure|BJ314H17HC]] on $X$ and $\mu^*(A) = \mu_0(A)$ for all $A \in \mathcal{A}$.
2. Every set of $\sigma(\mathcal{A})$, the [[sigma-algebra|0SV8K0TDGY]] generated by $\mathcal{A}$, is Carathéodory measurable for $\mu^*$.
   On $\sigma(\mathcal{A})$, the map $\mu^*$ is a [[measure|PR11BZQQPG]] that extends $\mu_0$.
3. If $\mu_0$ is [[sigma-finite|WEBBT18D4X]], that is, $X = \bigcup_n X_n$ with $X_n \in \mathcal{A}$ and $\mu_0(X_n) < \infty$, then this is the only measure on $\sigma(\mathcal{A})$ that extends $\mu_0$.

## Remarks

Clause 1 uses countable additivity of $\mu_0$: a countable cover of $A \in \mathcal{A}$ by sets of $\mathcal{A}$ can be made disjoint inside $A$, and then its total is at least $\mu_0(A)$.
For clause 2, a set $E \in \mathcal{A}$ splits each $A_n$ of a cover additively, so it splits every $B$; the Carathéodory-measurable sets form a $\sigma$-algebra, so they contain $\sigma(\mathcal{A})$.
Clause 3 follows from the $\pi$-$\lambda$ theorem, applied on each $X_n$.
Example: the finite unions of intervals $(a, b] \cap (0, 1]$ form an algebra on $(0, 1]$, length is a premeasure on it, and the theorem gives Lebesgue measure on the Borel subsets of $(0, 1]$.
The measure $\mu^*$ is in fact defined on all Carathéodory-measurable sets, which may be more than $\sigma(\mathcal{A})$; there it is a complete measure.

## Explanations

### L1 human

Start with sizes for a family of simple sets that is closed under finite unions and complements.
To size any other set, cover it by countably many simple sets as cheaply as possible.
The theorem says that this cheapest cover cost is a true measure on all sets that can be built from the simple ones by countable operations.

### L2 human

This is the algebra form of [[Carathéodory's extension theorem|Q8EXG7EKYW]], with an explicit formula for the extension.
In probability it is used to build measures on sequence spaces and, together with a compactness argument, the measures of the Kolmogorov extension theorem.
The formula for $\mu^*$ also shows that every measurable set can be approximated in measure by countable unions of sets of $\mathcal{A}$.