# Teorema de extensão de Carathéodory In measure theory, Teorema de extensão 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. Consequentemente, 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, por exemplo, 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. Conteúdo 1 Introductory statement 1.1 Comentários 2 Semi-ring and ring 2.1 Definições 2.2 Propriedades 2.3 Motivação 2.4 Exemplo 3 Declaração do teorema 3.1 Esboço de prova 4 Examples of non-uniqueness of extension 4.1 Via the counting measure 4.2 Via rationals 4.3 Via Fubini's theorem 5 Veja também 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. Um mais curto, simpler statement is as follows. Neste formulário, it is often called the Hahn–Kolmogorov theorem.

Deixar {estilo de exibição Sigma _{0}} be an algebra of subsets of a set {displaystyle X.} Consider a set function {mostre o estilo dele _{0}:Sigma _{0}para [0,infty ]} which is finitely additive, significa que {mostre o estilo dele _{0}deixei(copo grande _{n=1}^{N}UMA_{n}certo)=soma _{n=1}^{N}dentro _{0}(UMA_{n})} for any positive integer {estilo de exibição N} e {estilo de exibição A_{1},UMA_{2},ldots ,UMA_{N}} disjoint sets in {estilo de exibição Sigma _{0}.} Assume that this function satisfies the stronger sigma additivity assumption {mostre o estilo dele _{0}deixei(copo grande _{n=1}^{infty }UMA_{n}certo)=soma _{n=1}^{infty }dentro _{0}(UMA_{n})} for any disjoint family {estilo de exibição {UMA_{n}:nin mathbb {N} }} de elementos de {estilo de exibição Sigma _{0}} de tal modo que {displaystyle cup _{n=1}^{infty }UMA_{n}in Sigma _{0}.} (Functions {mostre o estilo dele _{0}} obeying these two properties are known as pre-measures.) Então, {mostre o estilo dele _{0}} extends to a measure defined on the {estilo de exibição sigma } -algebra {displaystyle Sigma } generated by {estilo de exibição Sigma _{0}} ; isso é, there exists a measure {mostre o estilo dele :Sigma to [0,infty ]} such that its restriction to {estilo de exibição Sigma _{0}} coincides with {mostre o estilo dele _{0}.} Se {mostre o estilo dele _{0}} é {estilo de exibição sigma } -finito, 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 {mostre o estilo dele _{0}} from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (E se {mostre o estilo dele _{0}} é {estilo de exibição sigma } -finito), 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 {estilo de exibição Omega ,} we call a family {estilo de exibição {matemática {S}}} of subsets of {estilo de exibição Omega } a semi-ring of sets if it has the following properties: {displaystyle varnothing in {matemática {S}}} Para todos {estilo de exibição A,Bin {matemática {S}},} temos {displaystyle Acap Bin {matemática {S}}} (closed under pairwise intersections) Para todos {estilo de exibição A,Bin {matemática {S}},} there exist disjoint sets {estilo de exibição K_{eu}dentro {matemática {S}},i=1,2,ldots ,n,} de tal modo que {displaystyle Asetminus B=bigcup _{i=1}^{n}K_{eu}} (relative complements can be written as finite disjoint unions).

The first property can be replaced with {estilo de exibição {matemática {S}}neq varnothing } desde {displaystyle Ain {matemática {S}}implies Asetminus A=varnothing in {matemática {S}}.} With the same notation, we call a family {estilo de exibição {matemática {R}}} of subsets of {estilo de exibição Omega } a ring of sets if it has the following properties: {displaystyle varnothing in {matemática {R}}} Para todos {estilo de exibição A,Bin {matemática {R}},} temos {displaystyle Acup Bin {matemática {R}}} (closed under pairwise unions) Para todos {estilo de exibição A,Bin {matemática {R}},} temos {displaystyle Asetminus Bin {matemática {R}}} (closed under relative complements).

Desta forma, any ring on {estilo de exibição Omega } is also a semi-ring.

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

Properties Arbitrary (possibly uncountable) intersections of rings on {estilo de exibição Omega } are still rings on {estilo de exibição Omega .} Se {estilo de exibição A} is a non-empty subset of the powerset {estilo de exibição {matemática {P}}(Ómega )} do {estilo de exibição Omega ,} then we define the ring generated by {estilo de exibição A} (noted {estilo de exibição R(UMA)} ) as the intersection of all rings containing {displaystyle A.} It is straightforward to see that the ring generated by {estilo de exibição A} is the smallest ring containing {displaystyle A.} For a semi-ring {estilo de exibição S,} the set of all finite unions of sets in {estilo de exibição S} is the ring generated by {estilo de exibição S:} {estilo de exibição R(S)= esquerda{UMA:A=bigcup _{i=1}^{n}UMA_{eu},UMA_{eu}in Sright}} (One can show that {estilo de exibição R(S)} is equal to the set of all finite disjoint unions of sets in {estilo de exibição S} ). A content {mostre o estilo dele } defined on a semi-ring {estilo de exibição S} can be extended on the ring generated by {displaystyle S.} Such an extension is unique. The extended content can be written: {mostre o estilo dele (UMA)=soma _{i=1}^{n}dentro (UMA_{eu})} por {displaystyle A=bigcup _{i=1}^{n}UMA_{eu},} with the {estilo de exibição A_{eu}em S} disjoint.

Além disso, it can be proved that {mostre o estilo dele } is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on {estilo de exibição R(S)} that extends the pre-measure on {estilo de exibição 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 {estilo de exibição S} (for example Stieltjes measures), which can then be extended to a pre-measure on {estilo de exibição 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). Na realidade, 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 (moreover it allows us to give an explicit representation of the smallest ring containing some semi-ring).

Example Think about the subset of {estilo de exibição {matemática {P}}(mathbb {R} )} defined by the set of all half-open intervals {estilo de exibição [uma,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.

Enunciado do teorema Seja {estilo de exibição R} be a ring of sets on {estilo de exibição X} e deixar {mostre o estilo dele :Rto [0,+infty ]} be a pre-measure on {estilo de exibição R,} meaning that for all sets {displaystyle Ain R} for which there exists a countable decomposition {displaystyle A=bigcup _{i=1}^{infty }UMA_{eu}} in disjoint sets {estilo de exibição A_{1},UMA_{2},ldots in R,} temos {mostre o estilo dele (UMA)=soma _{i=1}^{infty }dentro (UMA_{eu}).} Deixar {estilo de exibição sigma (R)} be the {estilo de exibição sigma } -algebra generated by {estilo de exibição R.} The pre-measure condition is a necessary condition for {mostre o estilo dele } to be the restriction to {estilo de exibição R} of a measure on {estilo de exibição sigma (R).} The Carathéodory's extension theorem states that it is also sufficient, isso é, there exists a measure {displaystyle mu ^{melhor }:sigma (R)para [0,+infty ]} de tal modo que {displaystyle mu ^{melhor }} is an extension of {mostre o estilo dele ;} isso é, {displaystyle mu ^{melhor }{big vert }_{R}=mu .} Além disso, E se {mostre o estilo dele } é {estilo de exibição sigma } -finite then the extension {displaystyle mu ^{melhor }} é único (and also {estilo de exibição sigma } -finito). Proof sketch First extend {mostre o estilo dele } to an outer measure {displaystyle mu ^{*}} on the power set {estilo de exibição 2 ^{X}} do {estilo de exibição X} por {displaystyle mu ^{*}(T)=inf left{soma _{n}mu left(S_{n}certo):Tsubseteq cup _{n}S_{n}{texto{ com }}S_{1},S_{2},ldots in Rright}} and then restrict it to the set {estilo de exibição {matemática {B}}} do {displaystyle mu ^{*}} -measurable sets (isso é, Carathéodory-measurable sets), which is the set of all {displaystyle Msubseteq X} de tal modo que {displaystyle mu ^{*}(S)=mu ^{*}(Scap M)+mu ^{*}(Scap M^{matemática {c} })} para cada {displaystyle Ssubseteq X.} It is a {estilo de exibição sigma } -algebra, e {displaystyle mu ^{*}} é {estilo de exibição sigma } -additive on it, by the Caratheodory lemma.

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

For uniqueness, take any other extension {estilo de exibição não } so it remains to show that {displaystyle nu =mu ^{*}.} Por {estilo de exibição sigma } -additivity, uniqueness can be reduced to the case where {mostre o estilo dele (X)} é finito, which will now be assumed.

Now we could concretely prove {displaystyle nu =mu ^{*}} sobre {estilo de exibição sigma (R)} by using the Borel hierarchy of {estilo de exibição R,} e desde {displaystyle nu =mu ^{*}} at the base level, we can use well-ordered induction to reach the level of {displaystyle omega _{1},} the level of {estilo de exibição 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 [uma,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 {estilo de exibição mathbb {Q} } of the form {estilo de exibição [uma,b)} , Onde {estilo de exibição a,sou mathbb {Q} } .

Deixar {estilo de exibição X} be {estilo de exibição mathbb {Q} boné [0,1)} e deixar {estilo de exibição Sigma _{0}} be the algebra of all finite unions of rational closed-open intervals contained in {estilo de exibição mathbb {Q} boné [0,1)} . It is easy to prove that {estilo de exibição Sigma _{0}} é, na verdade, an algebra. It is also easy to see that the cardinal of every non-empty set in {estilo de exibição Sigma _{0}} é {displaystyle aleph _{0}} .

Deixar {mostre o estilo dele _{0}} be the counting set function ( {estilo de exibição #} ) defined in {estilo de exibição Sigma _{0}} . É claro que {mostre o estilo dele _{0}} is finitely additive and {estilo de exibição sigma } -additive in {estilo de exibição Sigma _{0}} . Since every non-empty set in {estilo de exibição Sigma _{0}} is infinite, então, for every non-empty set {displaystyle Ain Sigma _{0}} , {mostre o estilo dele _{0}(UMA)=+infty } Agora, deixar {displaystyle Sigma } be the {estilo de exibição sigma } -algebra generated by {estilo de exibição Sigma _{0}} . It is easy to see that {displaystyle Sigma } is the Borel {estilo de exibição sigma } -algebra of subsets of {estilo de exibição X} , and both {estilo de exibição #} e {estilo de exibição 2#} are measures defined on {displaystyle Sigma } and both are extensions of {mostre o estilo dele _{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(UMA)card(B). This has a very large number of different extensions to a measure; por exemplo: 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 {estilo de exibição 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. No entanto, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) quem provou isso em 1924." Paul Loya (página 33). ^ Klenke, Achim (2014). Probability Theory. Universittext. p. Teorema 1.53. doi:10.1007/978-1-4471-5361-0. ISBN 978-1-4471-5360-3. ^ Vaillant, Noel. "Caratheodory's Extension" (PDF). Probability.net. Teorema 4. ^ Ash, Robert B. (1999). Probability and Measure Theory (2ª edição). Imprensa Acadêmica. p. 19. ISBN 0-12-065202-1.

This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, que está licenciado sob a Licença Creative Commons Atribuição/Compartilhamento.

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