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# 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

:$mu_0colon Sigma_0 omathbb\left\{R\right\}cup \left\{infty\right\}$

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 $\left\{A_n:nin mathbb\left\{N\right\}\right\}$ of elements of $Sigma_0$ such that $cup_\left\{n=1\right\}^infty A_nin Sigma_0$. 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

:$mucolonSigma o mathbb\left\{R\right\}cup\left\{infty\right\}$

such that its restriction to $Sigma_0$ coincides with $mu_0.$

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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