In mathematics, the Hahn-Kolmogorov theorem characterizes when a finitely additive function with non-negative (possibly infinite) values can be extended to a bona fide measure. It is named after the Austrian mathematician Hans Hahn and the Russian/Soviet mathematician Andrey Kolmogorov.
Statement of the theorem
Let <math>\Sigma_0</math> be an algebra of subsets of a set <math>X.</math> Consider a function
- <math>\mu_0\colon \Sigma_0 \to\mathbb{R}\cup \{\infty\}</math>
which is finitely additive, meaning that
- <math>\mu_0(\bigcup_{n=1}^N A_n)=\sum_{n=1}^N \mu_0(A_n)</math>
for any positive integer N and <math>A_1, A_2, ..., A_N</math> disjoint sets in <math>\Sigma_0</math>. Assume that this function satisfies the stronger sigma additivity assumption
- <math> \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>
for any disjoint family <math>\{A_n:n\in \mathbb{N}\}</math> of elements of <math>\Sigma_0</math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>. Then, <math>\mu_0</math> extends uniquely to a measure defined on the sigma-algebra <math>\Sigma</math> generated by <math>\Sigma_0</math>; i.e., there exists a unique measure
- <math>\mu\colon\Sigma\to \mathbb{R}\cup\{\infty\}</math>
such that its restriction to <math>\Sigma_0</math> coincides with <math>\mu_0.</math>
Comments
This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending <math>\mu_0</math> from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique, and moreover that it does not fail to satisfy the sigma-additivity of the original function.
This article incorporates material from Hahn-Kolmogorov theorem on PlanetMath, which is licensed under the GFDL.
