Théorème d'extension de Carathéodory

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In measure theory, Théorème d'extension de Carathéodory (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. Par conséquent, 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, par exemple, 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] Contenu 1 Introductory statement 1.1 commentaires 2 Semi-ring and ring 2.1 Définitions 2.2 Propriétés 2.3 Motivation 2.4 Exemple 3 Énoncé du théorème 3.1 Esquisse d'épreuve 4 Examples of non-uniqueness of extension 4.1 Via the counting measure 4.2 Via rationals 4.3 Via Fubini's theorem 5 Voir également 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. Un plus court, simpler statement is as follows. Sous cette forme, it is often called the Hahn–Kolmogorov theorem.

Laisser {style d'affichage Sigma _{0}} be an algebra of subsets of a set {style d'affichage X.} Consider a set function {style d'affichage lui _{0}:Sigma _{0}à [0,infime ]} which is finitely additive, qui veut dire {style d'affichage lui _{0}la gauche(grande tasse _{n=1}^{N}UN_{n}droit)=somme _{n=1}^{N}dans _{0}(UN_{n})} for any positive integer {displaystyle N} et {style d'affichage A_{1},UN_{2},ldots ,UN_{N}} disjoint sets in {style d'affichage Sigma _{0}.} Assume that this function satisfies the stronger sigma additivity assumption {style d'affichage lui _{0}la gauche(grande tasse _{n=1}^{infime }UN_{n}droit)=somme _{n=1}^{infime }dans _{0}(UN_{n})} for any disjoint family {style d'affichage {UN_{n}:nin mathbb {N} }} d'éléments de {style d'affichage Sigma _{0}} tel que {displaystyle cup _{n=1}^{infime }UN_{n}in Sigma _{0}.} (Functions {style d'affichage lui _{0}} obeying these two properties are known as pre-measures.) Alors, {style d'affichage lui _{0}} extends to a measure defined on the {style d'affichage sigma } -algebra {displaystyle Sigma } generated by {style d'affichage Sigma _{0}} ; C'est, there exists a measure {style d'affichage lui :Sigma to [0,infime ]} such that its restriction to {style d'affichage Sigma _{0}} coincides with {style d'affichage lui _{0}.} Si {style d'affichage lui _{0}} est {style d'affichage sigma } -fini, 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 {style d'affichage lui _{0}} from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (si {style d'affichage lui _{0}} est {style d'affichage sigma } -fini), 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 {style d'affichage Omega ,} we call a family {style d'affichage {mathématique {S}}} of subsets of {style d'affichage Omega } a semi-ring of sets if it has the following properties: {displaystyle varnothing in {mathématique {S}}} Pour tous {style d'affichage A,Bin {mathématique {S}},} Nous avons {displaystyle Acap Bin {mathématique {S}}} (closed under pairwise intersections) Pour tous {style d'affichage A,Bin {mathématique {S}},} there exist disjoint sets {style d'affichage K_{je}dans {mathématique {S}},i=1,2,ldots ,n,} tel que {displaystyle Asetminus B=bigcup _{je=1}^{n}K_{je}} (relative complements can be written as finite disjoint unions).

The first property can be replaced with {style d'affichage {mathématique {S}}neq varnothing } puisque {displaystyle Ain {mathématique {S}}implies Asetminus A=varnothing in {mathématique {S}}.} With the same notation, we call a family {style d'affichage {mathématique {R}}} of subsets of {style d'affichage Omega } a ring of sets if it has the following properties: {displaystyle varnothing in {mathématique {R}}} Pour tous {style d'affichage A,Bin {mathématique {R}},} Nous avons {displaystyle Acup Bin {mathématique {R}}} (closed under pairwise unions) Pour tous {style d'affichage A,Bin {mathématique {R}},} Nous avons {displaystyle Asetminus Bin {mathématique {R}}} (closed under relative complements).

Ainsi, any ring on {style d'affichage Omega } is also a semi-ring.

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

Properties Arbitrary (possibly uncountable) intersections of rings on {style d'affichage Omega } are still rings on {style d'affichage Omega .} Si {style d'affichage A} is a non-empty subset of the powerset {style d'affichage {mathématique {P}}(Oméga )} de {style d'affichage Omega ,} then we define the ring generated by {style d'affichage A} (noted {style d'affichage R(UN)} ) as the intersection of all rings containing {displaystyle A.} It is straightforward to see that the ring generated by {style d'affichage A} is the smallest ring containing {displaystyle A.} For a semi-ring {style d'affichage S,} the set of all finite unions of sets in {style d'affichage S} is the ring generated by {style d'affichage S:} {style d'affichage R(S)=gauche{UN:A=bigcup _{je=1}^{n}UN_{je},UN_{je}in Sright}} (One can show that {style d'affichage R(S)} is equal to the set of all finite disjoint unions of sets in {style d'affichage S} ). A content {style d'affichage lui } defined on a semi-ring {style d'affichage S} can be extended on the ring generated by {displaystyle S.} Such an extension is unique. The extended content can be written: {style d'affichage lui (UN)=somme _{je=1}^{n}dans (UN_{je})} pour {displaystyle A=bigcup _{je=1}^{n}UN_{je},} with the {style d'affichage A_{je}en S} disjoint.

en outre, it can be proved that {style d'affichage lui } is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on {style d'affichage R(S)} that extends the pre-measure on {style d'affichage 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 {style d'affichage S} (for example Stieltjes measures), which can then be extended to a pre-measure on {style d'affichage 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). Réellement, 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 {style d'affichage {mathématique {P}}(mathbb {R} )} defined by the set of all half-open intervals {style d'affichage [un,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.

Énoncé du théorème Soit {style d'affichage R} be a ring of sets on {style d'affichage X} et laissez {style d'affichage lui :Rto [0,+infime ]} be a pre-measure on {style d'affichage R,} meaning that for all sets {displaystyle Ain R} for which there exists a countable decomposition {displaystyle A=bigcup _{je=1}^{infime }UN_{je}} in disjoint sets {style d'affichage A_{1},UN_{2},ldots in R,} Nous avons {style d'affichage lui (UN)=somme _{je=1}^{infime }dans (UN_{je}).} Laisser {style d'affichage sigma (R)} be the {style d'affichage sigma } -algebra generated by {style d'affichage R.} The pre-measure condition is a necessary condition for {style d'affichage lui } to be the restriction to {style d'affichage R} of a measure on {style d'affichage sigma (R).} The Carathéodory's extension theorem states that it is also sufficient,[3] C'est, there exists a measure {displaystyle mu ^{prime }:sigma (R)à [0,+infime ]} tel que {displaystyle mu ^{prime }} is an extension of {style d'affichage lui ;} C'est, {displaystyle mu ^{prime }{big vert }_{R}=dans .} En outre, si {style d'affichage lui } est {style d'affichage sigma } -finite then the extension {displaystyle mu ^{prime }} est unique (and also {style d'affichage sigma } -fini).[4] Proof sketch First extend {style d'affichage lui } to an outer measure {displaystyle mu ^{*}} on the power set {style d'affichage 2 ^{X}} de {style d'affichage X} par {displaystyle mu ^{*}(J)=inf left{somme _{n}mu left(S_{n}droit):Tsubseteq cup _{n}S_{n}{texte{ avec }}S_{1},S_{2},ldots in Rright}} and then restrict it to the set {style d'affichage {mathématique {B}}} de {displaystyle mu ^{*}} -measurable sets (C'est, Carathéodory-measurable sets), which is the set of all {displaystyle Msubseteq X} tel que {displaystyle mu ^{*}(S)=mu ^{*}(Scap M)+mu ^{*}(Scap M^{mathrm {c} })} pour chaque {displaystyle Ssubseteq X.} It is a {style d'affichage sigma } -algebra, et {displaystyle mu ^{*}} est {style d'affichage sigma } -additive on it, by the Caratheodory lemma.

It remains to check that {style d'affichage {mathématique {B}}} contains {style d'affichage R.} C'est-à-dire, to verify that every set in {style d'affichage R} est {displaystyle mu ^{*}} -measurable. This is done by basic measure theory techniques of dividing and adding up sets.

For uniqueness, take any other extension {style d'affichage non } so it remains to show that {displaystyle nu =mu ^{*}.} Par {style d'affichage sigma } -additivity, uniqueness can be reduced to the case where {style d'affichage lui (X)} est fini, which will now be assumed.

Now we could concretely prove {displaystyle nu =mu ^{*}} sur {style d'affichage sigma (R)} by using the Borel hierarchy of {style d'affichage R,} et depuis {displaystyle nu =mu ^{*}} at the base level, we can use well-ordered induction to reach the level of {displaystyle omega _{1},} the level of {style d'affichage 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 [un,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 {style d'affichage mathbb {Q} } of the form {style d'affichage [un,b)} , où {style d'affichage a,suis mathbb {Q} } .

Laisser {style d'affichage X} être {style d'affichage mathbb {Q} cap [0,1)} et laissez {style d'affichage Sigma _{0}} be the algebra of all finite unions of rational closed-open intervals contained in {style d'affichage mathbb {Q} cap [0,1)} . It is easy to prove that {style d'affichage Sigma _{0}} est, En fait, an algebra. It is also easy to see that the cardinal of every non-empty set in {style d'affichage Sigma _{0}} est {displaystyle aleph _{0}} .

Laisser {style d'affichage lui _{0}} be the counting set function ( {style d'affichage #} ) defined in {style d'affichage Sigma _{0}} . Il est clair que {style d'affichage lui _{0}} is finitely additive and {style d'affichage sigma } -additive in {style d'affichage Sigma _{0}} . Since every non-empty set in {style d'affichage Sigma _{0}} is infinite, alors, for every non-empty set {displaystyle Ain Sigma _{0}} , {style d'affichage lui _{0}(UN)=+infty } À présent, laisser {displaystyle Sigma } be the {style d'affichage sigma } -algebra generated by {style d'affichage Sigma _{0}} . It is easy to see that {displaystyle Sigma } is the Borel {style d'affichage sigma } -algebra of subsets of {style d'affichage X} , and both {style d'affichage #} et {style d'affichage 2#} are measures defined on {displaystyle Sigma } and both are extensions of {style d'affichage lui _{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(UN)card(B). This has a very large number of different extensions to a measure; par exemple: 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 {style d'affichage entier _{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. Cependant, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) qui l'a prouvé dans 1924." Paul Loya (page 33). ^ Klenke, Achim (2014). Probability Theory. Universitext. p. Théorème 1.53. est ce que je:10.1007/978-1-4471-5361-0. ISBN 978-1-4471-5360-3. ^ Vaillant, Noel. "Caratheodory's Extension" (PDF). Théorème 4. ^ Ash, Robert B. (1999). Probability and Measure Theory (2sd éd.). Presse académique. p. 19. ISBN 0-12-065202-1.

This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, qui est sous licence Creative Commons Attribution/Share-Alike License.

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