Carathéodory's extension theorem

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 HahnKolmogorov extension theorem.[1]

Introductory statement

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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 sigma additive, meaning that   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

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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

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Definitions

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For a given set   we call a family   of subsets of   a semi-ring of sets 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 ring of sets 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

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  • 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

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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).

Example

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Think about the subset of   defined by the set of all half-open intervals   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

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Let   be a ring of sets on   and let   be a pre-measure on   meaning that   and for all sets   for which there exists a countable decomposition   in disjoint sets   we have  

Let   be the  -algebra generated by   The pre-measure condition is a necessary condition for   to be the restriction to   of a measure on   The Carathéodory's extension theorem states that it is also sufficient,[3] that is, there exists a measure   such that   is an extension of   that is,   Moreover, if   is  -finite then the extension   is unique (and also  -finite).[4]

Proof sketch

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First extend   to an outer measure   on the power set   of   by   and then restrict it to the set   of  -measurable sets (that is, Carathéodory-measurable sets), which is the set of all   such that   for every     is a  -algebra, and   is  -additive on it, by the Caratheodory lemma.

It remains to check that   contains   That is, to verify that every set in   is  -measurable. This is done by basic measure theory techniques of dividing and adding up sets.

For uniqueness, take any other extension   so it remains to show that   By  -additivity, uniqueness can be reduced to the case where   is finite, which will now be assumed.

Now we could concretely prove   on   by using the Borel hierarchy of   and since   at the base level, we can use well-ordered induction to reach the level of   the level of  

Examples of non-uniqueness of extension

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There can be more than one extension of a pre-measure to the generated σ-algebra, if the pre-measure is not  -finite, even if the extensions themselves are  -finite (see example "Via rationals" below).

Via the counting measure

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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. Another extension is given by the counting measure.

Via rationals

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This example is a more detailed variation of the above. The rational closed-open interval is any subset of   of the form  , where  .

Let   be   and let   be the algebra of all finite unions of rational closed-open intervals contained in  . It is easy to prove that   is, in fact, an algebra. It is also easy to see that the cardinal of every non-empty set in   is  .

Let   be the counting set function ( ) defined in  . It is clear that   is finitely additive and  -additive in  . Since every non-empty set in   is infinite, then, for every non-empty set  ,  

Now, let   be the  -algebra generated by  . It is easy to see that   is the  -algebra of all subsets of  , and both   and   are measures defined on   and both are extensions of  . Note that, in this case, the two extensions are  -finite, because   is countable.

Via Fubini's theorem

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Another example is closely related to the failure of some forms of Fubini's theorem for spaces that are not σ-finite. Suppose that   is the unit interval with Lebesgue measure and   is the unit interval with the discrete counting measure. Let the ring   be generated by products   where   is Lebesgue measurable and   is any subset, and give this set the measure  . This has a very large number of different extensions to a measure; for example:

  • The measure of a subset is the sum of the measures of its horizontal sections. This is the smallest possible extension. Here the diagonal has measure 0.
  • The measure of a subset is   where   is the number of points of the subset with given  -coordinate. The diagonal has measure 1.
  • The Carathéodory extension, which is the largest possible extension. Any subset of finite measure is contained in some union of a countable number of horizontal lines. In particular the diagonal has measure infinity.

See also

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  • Outer measure: the proof of Carathéodory's extension theorem is based upon the outer measure concept.
  • Loeb measures, constructed using Carathéodory's extension theorem.

References

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  1. ^ Quoting Paul Loya: "Warning: I've seen the following theorem called the Carathéodory extension theorem, the Carathéodory-Fréchet extension theorem, the Carathéodory-Hopf extension theorem, the Hopf extension theorem, the Hahn-Kolmogorov extension theorem, and many others that I can't remember! We shall simply call it Extension Theorem. However, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) who proved it in 1924." Paul Loya (page 33).
  2. ^ Klenke, Achim (2014). Probability Theory. Universitext. p. Theorem 1.53. doi:10.1007/978-1-4471-5361-0. ISBN 978-1-4471-5360-3.
  3. ^ Vaillant, Noel. "Caratheodory's Extension" (PDF). Probability.net. Theorem 4.
  4. ^ Ash, Robert B. (1999). Probability and Measure Theory (2nd ed.). Academic Press. p. 19. ISBN 0-12-065202-1.

This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.