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. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.
The theorem is also sometimes known as the Carathéodory–Fréchet extension theorem, the Carathéodory–Hopf extension theorem, the Hopf extension theorem and the Hahn–Kolmogorov extension theorem.
Introductory statement
Several very similar statements of the theorem can be given. A slightly more involved one, based on semi-rings of sets, is given further down below. A shorter, simpler statement is as follows. In this form, it is often called the Hahn–Kolmogorov theorem.
Let be an algebra of subsets of a set Consider a set function
which is finitely additive, meaning that
for any positive integer 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 -algebra generated by ; that is, there exists a measure
such that its restriction to coincides with
If is -finite, then the extension is unique.
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.
Semi-ring and ring
Definitions
For a given set we call a family of subsets of a if it has the following properties:
For all we have (closed under pairwise intersections) For all there exists a finite number of disjoint sets such that (relative complements can be written as finite disjoint unions).
The first property can be replaced with since
With the same notation, we call a family of subsets of a if it has the following properties:
For all we have (closed under pairwise unions) For all we have (closed under relative complements).
Thus, any ring on is also a semi-ring.
Sometimes, the following constraint is added in the measure theory context:
is the disjoint union of a countable family of sets in
A field of sets (respectively, a semi-field) is a ring (respectively, a semi-ring) that also contains as one of its elements.
Properties
Arbitrary (possibly uncountable) intersections of rings on are still rings on
If is a non-empty subset of the powerset of then we define the ring generated by (noted ) as the intersection of all rings containing It is straightforward to see that the ring generated by is the smallest ring containing For a semi-ring the set of all finite unions of sets in is the ring generated by (One can show that is equal to the set of all finite disjoint unions of sets in ). A content defined on a semi-ring can be extended on the ring generated by Such an extension is unique. The extended content can be written: for with the disjoint. In addition, it can be proved that is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on that extends the pre-measure on is necessarily of this form.
Motivation
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.