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.

tatement of the theorem

Let $Sigma\_0$ be an algebra of subsets of a set $X.$ Consider a function

:$mu\_0colon\; Sigma\_0\; omathbb\{R\}cup\; \{infty\}$

which is "finitely additive", meaning that :

for any positive integer "N" and $A\_1,\; A\_2,\; dots,\; A\_N$ disjoint sets in $Sigma\_0$.

Assume that this function satisfies the stronger "sigma additivity" assumption

:

for any disjoint family $\{A\_n:nin\; mathbb\{N\}\}$ of elements of $Sigma\_0$ such that $cup\_\{n=1\}^infty\; A\_nin\; Sigma\_0$. 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

:$mucolonSigma\; o\; mathbb\{R\}cup\{infty\}$

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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