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 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 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 (if
is
-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
See also
This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.