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.
Let
Σ
0
be an algebra of subsets of a set
X
.
Consider a function
μ
0
:
Σ
0
→
[
0
,
∞
]
which is finitely additive, meaning that
μ
0
(
⋃
n
=
1
N
A
n
)
=
∑
n
=
1
N
μ
0
(
A
n
)
for any positive integer N and
A
1
,
A
2
,
…
,
A
N
disjoint sets in
Σ
0
.
Assume that this function satisfies the stronger sigma additivity assumption
μ
0
(
⋃
n
=
1
∞
A
n
)
=
∑
n
=
1
∞
μ
0
(
A
n
)
for any disjoint family
{
A
n
:
n
∈
N
}
of elements of
Σ
0
such that
∪
n
=
1
∞
A
n
∈
Σ
0
. (Functions
μ
0
obeying these two properties are known as pre-measures.) Then,
μ
0
extends to a measure defined on the sigma-algebra
Σ
generated by
Σ
0
; i.e., there exists a measure
μ
:
Σ
→
[
0
,
∞
]
such that its restriction to
Σ
0
coincides with
μ
0
.
If
μ
0
is
σ
-finite, then the extension is unique.
If
μ
0
is not
σ
-finite then the extension need not be unique, even if the extension itself is
σ
-finite.
Here is an example:
We call rational closed-open interval, any subset of
Q
of the form
[
a
,
b
)
, where
a
,
b
∈
Q
.
Let
X
be
Q
∩
[
0
,
1
)
and let
Σ
0
be the algebra of all finite union of rational closed-open intervals contained in
Q
∩
[
0
,
1
)
. It is easy to prove that
Σ
0
is, in fact, an algebra. It is also easy to see that every non-empty set in
Σ
0
is infinite.
Let
μ
0
be the counting set function (
#
) defined in
Σ
0
. It is clear that
μ
0
is finitely additive and
σ
-additive in
Σ
0
. Since every non-empty set in
Σ
0
is infinite, we have, for every non-empty set
A
∈
Σ
0
,
μ
0
(
A
)
=
+
∞
Now, let
Σ
be the
σ
-algebra generated by
Σ
0
. It is easy to see that
Σ
is the Borel
σ
-algebra of subsets of
X
, and both
#
and
2
#
are measures defined on
Σ
and both are extensions of
μ
0
.
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
μ
0
from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if
μ
0
is
σ
-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
