Hahn–Kolmogorov theorem


In mathematics, the Hahn–Kolmogorov theorem characterizes when a finitely additive function with non-negative 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 be an algebra of subsets of a set Consider a function
which is finitely additive, meaning that
for any positive integer N and disjoint sets in.
Assume that this function satisfies the stronger sigma additivity assumption
for any disjoint family of elements of such that . 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 the cardinal of every non-empty set in is.
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.

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 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.