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 Russia n/Soviet mathematician Andrey Kolmogorov.
Statement of the theorem
Let [math]\displaystyle{ \Sigma_0 }[/math] be an algebra of subsets of a set [math]\displaystyle{ X. }[/math] Consider a function
- [math]\displaystyle{ \mu_0\colon \Sigma_0 \to[0,\infty] }[/math]
which is finitely additive, meaning that
- [math]\displaystyle{ \mu_0\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu_0(A_n) }[/math]
for any positive integer N and [math]\displaystyle{ A_1, A_2, \dots, A_N }[/math] disjoint sets in [math]\displaystyle{ \Sigma_0 }[/math].
Assume that this function satisfies the stronger sigma additivity assumption
- [math]\displaystyle{ \mu_0\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu_0(A_n) }[/math]
for any disjoint family [math]\displaystyle{ \{A_n:n\in \mathbb{N}\} }[/math] of elements of [math]\displaystyle{ \Sigma_0 }[/math] such that [math]\displaystyle{ \cup_{n=1}^\infty A_n\in \Sigma_0 }[/math]. (Functions [math]\displaystyle{ \mu_0 }[/math] obeying these two properties are known as pre-measures.) Then, [math]\displaystyle{ \mu_0 }[/math] extends to a measure defined on the sigma-algebra [math]\displaystyle{ \Sigma }[/math] generated by [math]\displaystyle{ \Sigma_0 }[/math]; i.e., there exists a measure
- [math]\displaystyle{ \mu \colon \Sigma \to[0,\infty] }[/math]
such that its restriction to [math]\displaystyle{ \Sigma_0 }[/math] coincides with [math]\displaystyle{ \mu_0. }[/math]
If [math]\displaystyle{ \mu_0 }[/math] is [math]\displaystyle{ \sigma }[/math]-finite, then the extension is unique.
Non-uniqueness of the extension
If [math]\displaystyle{ \mu_0 }[/math] is not [math]\displaystyle{ \sigma }[/math]-finite then the extension need not be unique, even if the extension itself is [math]\displaystyle{ \sigma }[/math]-finite.
Here is an example:
We call rational closed-open interval, any subset of [math]\displaystyle{ \mathbb{Q} }[/math] of the form [math]\displaystyle{ [a,b) }[/math], where [math]\displaystyle{ a, b \in \mathbb{Q} }[/math].
Let [math]\displaystyle{ X }[/math] be [math]\displaystyle{ \mathbb{Q}\cap[0,1) }[/math] and let [math]\displaystyle{ \Sigma_0 }[/math] be the algebra of all finite union of rational closed-open intervals contained in [math]\displaystyle{ \mathbb{Q}\cap[0,1) }[/math]. It is easy to prove that [math]\displaystyle{ \Sigma_0 }[/math] is, in fact, an algebra. It is also easy to see that the cardinal of every non-empty set in [math]\displaystyle{ \Sigma_0 }[/math] is [math]\displaystyle{ \aleph_0 }[/math].
Let [math]\displaystyle{ \mu_0 }[/math] be the counting set function ([math]\displaystyle{ \# }[/math]) defined in [math]\displaystyle{ \Sigma_0 }[/math]. It is clear that [math]\displaystyle{ \mu_0 }[/math] is finitely additive and [math]\displaystyle{ \sigma }[/math]-additive in [math]\displaystyle{ \Sigma_0 }[/math]. Since every non-empty set in [math]\displaystyle{ \Sigma_0 }[/math] is infinite, we have, for every non-empty set [math]\displaystyle{ A\in\Sigma_0 }[/math], [math]\displaystyle{ \mu_0(A)=+\infty }[/math]
Now, let [math]\displaystyle{ \Sigma }[/math] be the [math]\displaystyle{ \sigma }[/math]-algebra generated by [math]\displaystyle{ \Sigma_0 }[/math]. It is easy to see that [math]\displaystyle{ \Sigma }[/math] is the Borel [math]\displaystyle{ \sigma }[/math]-algebra of subsets of [math]\displaystyle{ X }[/math], and both [math]\displaystyle{ \# }[/math] and [math]\displaystyle{ 2\# }[/math] are measures defined on [math]\displaystyle{ \Sigma }[/math] and both are extensions of [math]\displaystyle{ \mu_0 }[/math].
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 [math]\displaystyle{ \mu_0 }[/math] from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if [math]\displaystyle{ \mu_0 }[/math] is [math]\displaystyle{ \sigma }[/math]-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
See also
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