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 : $$mu_0(igcup_{n=1}^N A_n)=sum_{n=1}^N mu_0(A_n)

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

:$$mu_0(igcup_{n=1}^infty A_n) = sum_{n=1}^infty mu_0(A_n)

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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