## You are here

HomeHahn-Kolmogorov theorem

## Primary tabs

# Hahn-Kolmogorov theorem

Let $\mathscr{A}_{0}$ be an algebra of subsets of a set $X$. If a finitely additive measure $\mu_{0}\colon\mathscr{A}\to\mathbb{R}\cup\{\infty\}$ satisfies

$\mu_{0}(\bigcup_{{n=1}}^{\infty}A_{n})=\sum_{{n=1}}^{\infty}\mu_{0}(A_{n})$ |

for any disjoint family $\{A_{n}:n\in\mathbb{N}\}$ of elements of $\mathscr{A}_{0}$ such that $\cup_{{n=0}}^{\infty}A_{n}\in\mathscr{A}_{0}$, then $\mu_{0}$ extends to a measure defined on the $\sigma$-algebra $\mathscr{A}$ generated by $\mathscr{A}_{0}$; i.e. there exists a measure $\mu\colon\mathscr{A}\to\mathbb{R}\cup\{\infty\}$ such that its restriction to $\mathscr{A}_{0}$ coincides with $\mu_{0}$.

If $\mu_{0}$ is $\sigma$-finite, then the extension is unique.

Synonym:

Hahn-Kolmogorov extension theorem, Kolmogorov extension theorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

28A10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb