Carathéodory's extension theorem

In measure theory, Carathéodory's extension theorem (named after the Greek mathematician Constantin Carathéodory) states that any measure defined on a given ring R of subsets of a given set Ω can be extended to the σ-algebra generated by R, and this extension is unique if the measure is σ-finite. Consequently, any measure on a space 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 proves, for example, the existence of the Lebesgue measure.

Semi-ring and ring


For a given set Ω, we may define a semi-ring as a subset S of , the power set of Ω, which has the following properties:

(The first property can be replaced with "S is not empty" since A \ A = ∅ must be in S if A is in S).

With the same notation, we define a ring R as a subset of the power set of Ω which has the following properties:

Thus any ring on Ω is also a semi-ring.

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 Ω as one of its elements.


(One can show that R(S) is simply the set containing all finite unions of sets in S).

for , with the Ai in S.

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(S) that extends the pre-measure on S is necessarily of this form.


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 (for example Stieltjes measures), which can then be extended to a pre-measure on R(S), 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, the Carathéodory's extension theorem can be slightly generalized by replacing ring by semi-ring.[1]

The definition of semi-ring may seem a bit convoluted, but the following example shows why it is useful.


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 countably union of intervals is proved using Caratheodory's theorem.

Statement of the theorem

Let R be a ring on Ω and μ:R → [0, + ∞] be a pre-measure on R.

The Carathéodory's extension theorem states that[2] there exists a measure μ′:σ(R) → [0, + ∞] such that μ′ is an extension of μ. (That is, μ′|R = μ).

Here σ(R) is the σ-algebra generated by R.

If μ is σ-finite then the extension μ′ is unique (and also σ-finite).[3]


Non-uniqueness of extension

Here is some examples where there is more than one extension of a pre-measure to the σ-algebra generated by it.

For the first example, take the algebra generated by all half-open intervals [a,b) on the real, 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 μ(A)card(B). This has a very large number of different extensions to a measure; for example:

See also


  1. Klenke, Achim (2014). Probability Theory. p. Theorem 1.53. ISBN 978-1-4471-5360-3.
  2. Vaillant, Theorem 4
  3. Ash, p19
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