Hahn–Kolmogorov theorem

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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[edit]

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 . (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[edit]

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[edit]

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.

See also[edit]

This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.