Carathéodory's extension theorem


In measure theory, Carathéodory's extension theorem states that any pre-measure defined on a given ring R of subsets of a given set Ω can be extended to a measure on the σ-algebra 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.

Semi-ring and ring

Definitions

For a given set, we may define a semi-ring as a subset of, the power set of, which has the following properties:
The first property can be replaced with since.
With the same notation, we define a ring as a subset of the power set of which has the following properties:
Thus, any ring on is also a semi-ring.
Sometimes, the following constraint is added in the measure theory context:
A field of sets is a ring that also contains as one of its elements.

Properties

.
In addition, it can be proved that μ is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on R that extends the pre-measure on S 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 S, which can then be extended to a pre-measure on R, 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. Actually, Carathéodory's extension theorem can be slightly generalized by replacing ring by semi-field.
The definition of semi-ring may seem a bit convoluted, but the following example shows why it is useful.

Example

Think about the subset of defined by the set of all half-open intervals a, b) for a and b reals. This is a semi-ring, but not a ring. [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.

Statement of the theorem

Let be a ring on and let be a pre-measure on R, i.e. for all sets for which there exists a countable decomposition in disjoint sets, we have.
Let σ be the σ-algebra generated by R. The pre-measure condition is a necessary condition for to be the restriction to R of a measure on.
The Carathéodory's extension theorem states that it is also sufficient, i.e. there exists a measure such that is an extension of μ.. Moreover, if μ is σ-finite then the extension is unique.

Examples

Non-uniqueness of extension

Here are some examples where there is more than one extension of a pre-measure to the generated σ-algebra.
For the first example, 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 Caratheodory extension gives all non-empty sets measure infinity. Another extension is given by counting measure.
Here is a second example, closely related to the failure of some forms of Fubini's theorem for spaces that are not σ-finite.
Suppose that X is the unit interval with Lebesgue measure and Y is the unit interval with the discrete counting measure. Let the ring R be generated by products A×B where A is Lebesgue measurable and B is any subset, and give this set the measure μcard. This has a very large number of different extensions to a measure; for example: