Hahn–Kolmogorov theorem

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 $\Sigma _{0}$ be an algebra of subsets of a set $X.$ Consider a function

$\mu _{0}\colon \Sigma _{0}\to \mathbb {R} \cup \{\infty \}$

which is finitely additive, meaning that

$\mu _{0}(\bigcup _{n=1}^{N}A_{n})=\sum _{n=1}^{N}\mu _{0}(A_{n})$

for any positive integer N and $A_{1},A_{2},...,A_{N}$ disjoint sets in $\Sigma _{0}$.

Assume that this function satisfies the stronger sigma additivity assumption

$\mu _{0}(\bigcup _{n=1}^{\infty }A_{n})=\sum _{n=1}^{\infty }\mu _{0}(A_{n})$

for any disjoint family $\{A_{n}:n\in \mathbb {N} \}$ of elements of $\Sigma _{0}$ such that $\cup _{n=1}^{\infty }A_{n}\in \Sigma _{0}$. Then, $\mu _{0}$ extends uniquely to a measure defined on the sigma-algebra $\Sigma$ generated by $\Sigma _{0}$; i.e., there exists a unique measure

$\mu \colon \Sigma \to \mathbb {R} \cup \{\infty \}$

such that its restriction to $\Sigma _{0}$ coincides with $\mu _{0}.$

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 $\mu _{0}$ 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.

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Hahn-Kolmogorov theorem at PlanetMath.