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
which is finitely additive, meaning that
Assume that this function satisfies the stronger sigma additivity assumption
for any disjoint family of elements of such that . (Functions obeying these two properties are known as pre-measures.) Then, extends to a measure defined on the sigma-algebra generated by ; i.e., there exists a measure
such that its restriction to coincides with
If is -finite, then the extension is unique.
Non-uniqueness of the extension
If is not -finite then the extension need not be unique, even if the extension itself is -finite.
Here is an example:
We call rational closed-open interval, any subset of of the form , where .
Let be and let be the algebra of all finite union of rational closed-open intervals contained in . It is easy to prove that is, in fact, an algebra. It is also easy to see that every non-empty set in is infinite.
Let be the counting set function () defined in . It is clear that is finitely additive and -additive in . Since every non-empty set in is infinite, we have, for every non-empty set ,
Now, let be the -algebra generated by . It is easy to see that is the Borel -algebra of subsets of , and both and are measures defined on and both are extensions of .
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 from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if is -finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.