In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite.
Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers.
A slightly more involved one, based on semi-rings of sets, is given further down below.
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
-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.
Sometimes, the following constraint is added in the measure theory context: A field of sets (respectively, a semi-field) is a ring (respectively, a semi-ring) that also contains
In measure theory, we are not interested in semi-rings and rings themselves, but rather in σ-algebras generated by them.
The idea is that it is possible to build a pre-measure on a semi-ring
(for example Stieltjes measures), which can then be extended to a pre-measure on
which can finally be extended to a measure on a σ-algebra through Caratheodory's extension theorem.
As σ-algebras generated by semi-rings and rings are the same, the difference does not really matter (in the measure theory context at least).
Actually, Carathéodory's extension theorem can be slightly generalized by replacing ring by semi-field.
[2] The definition of semi-ring may seem a bit convoluted, but the following example shows why it is useful (moreover it allows us to give an explicit representation of the smallest ring containing some semi-ring).
defined by the set of all half-open intervals
Stieltjes measures are defined on intervals; the countable additivity on the semi-ring is not too difficult to prove because we only consider countable unions of intervals which are intervals themselves.
Proving it for arbitrary countable unions of intervals is accomplished using Caratheodory's theorem.
The Carathéodory's extension theorem states that it is also sufficient,[3] that is, there exists a measure
This is done by basic measure theory techniques of dividing and adding up sets.
Take the algebra generated by all half-open intervals [a,b) on the real line, and give such intervals measure infinity if they are non-empty.
The Carathéodory extension gives all non-empty sets measure infinity.
The rational closed-open interval is any subset of
be the algebra of all finite unions of rational closed-open intervals contained in
It is also easy to see that the cardinal of every non-empty set in
Another example is closely related to the failure of some forms of Fubini's theorem for spaces that are not σ-finite.
is the unit interval with Lebesgue measure and
is the unit interval with the discrete counting measure.
is any subset, and give this set the measure
This has a very large number of different extensions to a measure; for example: This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.