# 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 (http://planetmath.org/RestrictionOfAFunction) to $\mathscr{A}_{0}$ coincides with $\mu_{0}$.

If $\mu_{0}$ is $\sigma$-finite (http://planetmath.org/SigmaFinite), then the extension is unique.

Title Hahn-Kolmogorov theorem HahnKolmogorovTheorem 2013-03-22 14:03:10 2013-03-22 14:03:10 Koro (127) Koro (127) 7 Koro (127) Theorem msc 28A10 Hahn-Kolmogorov extension theorem Kolmogorov extension theorem