Carathéodory's extension theorem

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In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) 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.

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.[1] Contents 1 Introductory statement 1.1 Comments 2 Semi-ring and ring 2.1 Definitions 2.2 Properties 2.3 Motivation 2.4 Example 3 Statement of the theorem 3.1 Proof sketch 4 Examples of non-uniqueness of extension 4.1 Via the counting measure 4.2 Via rationals 4.3 Via Fubini's theorem 5 See also 6 References 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 {displaystyle Sigma _{0}} be an algebra of subsets of a set {displaystyle X.} Consider a set function {displaystyle mu _{0}:Sigma _{0}to [0,infty ]} which is finitely additive, meaning that {displaystyle mu _{0}left(bigcup _{n=1}^{N}A_{n}right)=sum _{n=1}^{N}mu _{0}(A_{n})} for any positive integer {displaystyle N} and {displaystyle A_{1},A_{2},ldots ,A_{N}} disjoint sets in {displaystyle Sigma _{0}.} Assume that this function satisfies the stronger sigma additivity assumption {displaystyle mu _{0}left(bigcup _{n=1}^{infty }A_{n}right)=sum _{n=1}^{infty }mu _{0}(A_{n})} for any disjoint family {displaystyle {A_{n}:nin mathbb {N} }} of elements of {displaystyle Sigma _{0}} such that {displaystyle cup _{n=1}^{infty }A_{n}in Sigma _{0}.} (Functions {displaystyle mu _{0}} obeying these two properties are known as pre-measures.) Then, {displaystyle mu _{0}} extends to a measure defined on the {displaystyle sigma } -algebra {displaystyle Sigma } generated by {displaystyle Sigma _{0}} ; that is, there exists a measure {displaystyle mu :Sigma to [0,infty ]} such that its restriction to {displaystyle Sigma _{0}} coincides with {displaystyle mu _{0}.} If {displaystyle mu _{0}} is {displaystyle sigma } -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 {displaystyle mu _{0}} from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if {displaystyle mu _{0}} is {displaystyle sigma } -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 {displaystyle Omega ,} we call a family {displaystyle {mathcal {S}}} of subsets of {displaystyle Omega } a semi-ring of sets if it has the following properties: {displaystyle varnothing in {mathcal {S}}} For all {displaystyle A,Bin {mathcal {S}},} we have {displaystyle Acap Bin {mathcal {S}}} (closed under pairwise intersections) For all {displaystyle A,Bin {mathcal {S}},} there exist disjoint sets {displaystyle K_{i}in {mathcal {S}},i=1,2,ldots ,n,} such that {displaystyle Asetminus B=bigcup _{i=1}^{n}K_{i}} (relative complements can be written as finite disjoint unions).

The first property can be replaced with {displaystyle {mathcal {S}}neq varnothing } since {displaystyle Ain {mathcal {S}}implies Asetminus A=varnothing in {mathcal {S}}.} With the same notation, we call a family {displaystyle {mathcal {R}}} of subsets of {displaystyle Omega } a ring of sets if it has the following properties: {displaystyle varnothing in {mathcal {R}}} For all {displaystyle A,Bin {mathcal {R}},} we have {displaystyle Acup Bin {mathcal {R}}} (closed under pairwise unions) For all {displaystyle A,Bin {mathcal {R}},} we have {displaystyle Asetminus Bin {mathcal {R}}} (closed under relative complements).

Thus, any ring on {displaystyle Omega } is also a semi-ring.

Sometimes, the following constraint is added in the measure theory context: {displaystyle Omega } is the disjoint union of a countable family of sets in {displaystyle {mathcal {S}}.} A field of sets (respectively, a semi-field) is a ring (respectively, a semi-ring) that also contains {displaystyle Omega } as one of its elements.

Properties Arbitrary (possibly uncountable) intersections of rings on {displaystyle Omega } are still rings on {displaystyle Omega .} If {displaystyle A} is a non-empty subset of the powerset {displaystyle {mathcal {P}}(Omega )} of {displaystyle Omega ,} then we define the ring generated by {displaystyle A} (noted {displaystyle R(A)} ) as the intersection of all rings containing {displaystyle A.} It is straightforward to see that the ring generated by {displaystyle A} is the smallest ring containing {displaystyle A.} For a semi-ring {displaystyle S,} the set of all finite unions of sets in {displaystyle S} is the ring generated by {displaystyle S:} {displaystyle R(S)=left{A:A=bigcup _{i=1}^{n}A_{i},A_{i}in Sright}} (One can show that {displaystyle R(S)} is equal to the set of all finite disjoint unions of sets in {displaystyle S} ). A content {displaystyle mu } defined on a semi-ring {displaystyle S} can be extended on the ring generated by {displaystyle S.} Such an extension is unique. The extended content can be written: {displaystyle mu (A)=sum _{i=1}^{n}mu (A_{i})} for {displaystyle A=bigcup _{i=1}^{n}A_{i},} with the {displaystyle A_{i}in S} disjoint.

In addition, it can be proved that {displaystyle mu } is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on {displaystyle R(S)} that extends the pre-measure on {displaystyle 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 {displaystyle S} (for example Stieltjes measures), which can then be extended to a pre-measure on {displaystyle 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, 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 Think about the subset of {displaystyle {mathcal {P}}(mathbb {R} )} defined by the set of all half-open intervals {displaystyle [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 {displaystyle R} be a ring of sets on {displaystyle X} and let {displaystyle mu :Rto [0,+infty ]} be a pre-measure on {displaystyle R,} meaning that for all sets {displaystyle Ain R} for which there exists a countable decomposition {displaystyle A=bigcup _{i=1}^{infty }A_{i}} in disjoint sets {displaystyle A_{1},A_{2},ldots in R,} we have {displaystyle mu (A)=sum _{i=1}^{infty }mu (A_{i}).} Let {displaystyle sigma (R)} be the {displaystyle sigma } -algebra generated by {displaystyle R.} The pre-measure condition is a necessary condition for {displaystyle mu } to be the restriction to {displaystyle R} of a measure on {displaystyle sigma (R).} The Carathéodory's extension theorem states that it is also sufficient,[3] that is, there exists a measure {displaystyle mu ^{prime }:sigma (R)to [0,+infty ]} such that {displaystyle mu ^{prime }} is an extension of {displaystyle mu ;} that is, {displaystyle mu ^{prime }{big vert }_{R}=mu .} Moreover, if {displaystyle mu } is {displaystyle sigma } -finite then the extension {displaystyle mu ^{prime }} is unique (and also {displaystyle sigma } -finite).[4] Proof sketch First extend {displaystyle mu } to an outer measure {displaystyle mu ^{*}} on the power set {displaystyle 2^{X}} of {displaystyle X} by {displaystyle mu ^{*}(T)=inf left{sum _{n}mu left(S_{n}right):Tsubseteq cup _{n}S_{n}{text{ with }}S_{1},S_{2},ldots in Rright}} and then restrict it to the set {displaystyle {mathcal {B}}} of {displaystyle mu ^{*}} -measurable sets (that is, Carathéodory-measurable sets), which is the set of all {displaystyle Msubseteq X} such that {displaystyle mu ^{*}(S)=mu ^{*}(Scap M)+mu ^{*}(Scap M^{mathrm {c} })} for every {displaystyle Ssubseteq X.} It is a {displaystyle sigma } -algebra, and {displaystyle mu ^{*}} is {displaystyle sigma } -additive on it, by the Caratheodory lemma.

It remains to check that {displaystyle {mathcal {B}}} contains {displaystyle R.} That is, to verify that every set in {displaystyle R} is {displaystyle mu ^{*}} -measurable. This is done by basic measure theory techniques of dividing and adding up sets.

For uniqueness, take any other extension {displaystyle nu } so it remains to show that {displaystyle nu =mu ^{*}.} By {displaystyle sigma } -additivity, uniqueness can be reduced to the case where {displaystyle mu (X)} is finite, which will now be assumed.

Now we could concretely prove {displaystyle nu =mu ^{*}} on {displaystyle sigma (R)} by using the Borel hierarchy of {displaystyle R,} and since {displaystyle nu =mu ^{*}} at the base level, we can use well-ordered induction to reach the level of {displaystyle omega _{1},} the level of {displaystyle sigma (R).} Examples of non-uniqueness of extension There can be more than one extension of a pre-measure to the generated σ-algebra, if the pre-measure is not sigma-finite.

Via the counting measure 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 This example is a more detailed variation of the above. The rational closed-open interval is any subset of {displaystyle mathbb {Q} } of the form {displaystyle [a,b)} , where {displaystyle a,bin mathbb {Q} } .

Let {displaystyle X} be {displaystyle mathbb {Q} cap [0,1)} and let {displaystyle Sigma _{0}} be the algebra of all finite unions of rational closed-open intervals contained in {displaystyle mathbb {Q} cap [0,1)} . It is easy to prove that {displaystyle Sigma _{0}} is, in fact, an algebra. It is also easy to see that the cardinal of every non-empty set in {displaystyle Sigma _{0}} is {displaystyle aleph _{0}} .

Let {displaystyle mu _{0}} be the counting set function ( {displaystyle #} ) defined in {displaystyle Sigma _{0}} . It is clear that {displaystyle mu _{0}} is finitely additive and {displaystyle sigma } -additive in {displaystyle Sigma _{0}} . Since every non-empty set in {displaystyle Sigma _{0}} is infinite, then, for every non-empty set {displaystyle Ain Sigma _{0}} , {displaystyle mu _{0}(A)=+infty } Now, let {displaystyle Sigma } be the {displaystyle sigma } -algebra generated by {displaystyle Sigma _{0}} . It is easy to see that {displaystyle Sigma } is the Borel {displaystyle sigma } -algebra of subsets of {displaystyle X} , and both {displaystyle #} and {displaystyle 2#} are measures defined on {displaystyle Sigma } and both are extensions of {displaystyle mu _{0}} .

Via Fubini's theorem Another example is 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: 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 {displaystyle int _{0}^{1}n(x)dx} where n(x) is the number of points of the subset with given x-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 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 ^ 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). ^ Klenke, Achim (2014). Probability Theory. Universitext. p. Theorem 1.53. doi:10.1007/978-1-4471-5361-0. ISBN 978-1-4471-5360-3. ^ Vaillant, Noel. "Caratheodory's Extension" (PDF). Theorem 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.

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