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Hahn–Kolmogorov theorem: Difference between revisions

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:<math> \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>
:<math> \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>


for any disjoint family <math>\{A_n:n\in \mathbb{N}\}</math> of elements of <math>\Sigma_0</math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>. Then,
for any disjoint family <math>\{A_n:n\in \mathbb{N}\}</math> of elements of <math>\Sigma_0</math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>. (Functions <math>\mu_0</math> obeying these two properties are known as [[pre-measures]].) Then,
<math>\mu_0</math> extends to a measure defined on the [[sigma-algebra]] <math>\Sigma</math> generated by <math>\Sigma_0</math>; i.e., there exists a measure
<math>\mu_0</math> extends to a measure defined on the [[sigma-algebra]] <math>\Sigma</math> generated by <math>\Sigma_0</math>; i.e., there exists a measure



Revision as of 21:56, 3 July 2011

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

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

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 set of all subset of , and both and are -finite 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 (if is -finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.

See also


Hahn–Kolmogorov theorem at PlanetMath.