# Teorema dell'estensione di Carathéodory

In measure theory, Teorema dell'estensione di 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. Di conseguenza, 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, Per esempio, 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] Contenuti 1 Introductory statement 1.1 Commenti 2 Semi-ring and ring 2.1 Definizioni 2.2 Proprietà 2.3 Motivazione 2.4 Esempio 3 Enunciato del teorema 3.1 Schizzo di 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 Guarda anche 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 più corto, simpler statement is as follows. In questa forma, it is often called the Hahn–Kolmogorov theorem.

Permettere {displaystyle Sigma _{0}} be an algebra of subsets of a set {stile di visualizzazione X.} Consider a set function {displaystyle lui _{0}:Sigma _{0}a [0,infty ]} which is finitely additive, intendendo che {displaystyle lui _{0}sinistra(tazza grande _{n=1}^{N}UN_{n}Giusto)=somma _{n=1}^{N}in _{0}(UN_{n})} for any positive integer {stile di visualizzazione N} e {stile di visualizzazione A_{1},UN_{2},ldot ,UN_{N}} disjoint sets in {displaystyle Sigma _{0}.} Assume that this function satisfies the stronger sigma additivity assumption {displaystyle lui _{0}sinistra(tazza grande _{n=1}^{infty }UN_{n}Giusto)=somma _{n=1}^{infty }in _{0}(UN_{n})} for any disjoint family {stile di visualizzazione {UN_{n}:nin mathbb {N} }} di elementi di {displaystyle Sigma _{0}} tale che {displaystyle cup _{n=1}^{infty }UN_{n}in Sigma _{0}.} (Functions {displaystyle lui _{0}} obeying these two properties are known as pre-measures.) Quindi, {displaystyle lui _{0}} extends to a measure defined on the {displaystyle sigma } -algebra {stile di visualizzazione Sigma } generated by {displaystyle Sigma _{0}} ; questo è, there exists a measure {displaystyle lui :Sigma to [0,infty ]} such that its restriction to {displaystyle Sigma _{0}} coincide con {displaystyle lui _{0}.} Se {displaystyle lui _{0}} è {displaystyle 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 {displaystyle lui _{0}} from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (Se {displaystyle lui _{0}} è {displaystyle 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 {stile di visualizzazione Omega ,} we call a family {stile di visualizzazione {matematico {S}}} of subsets of {stile di visualizzazione Omega } a semi-ring of sets if it has the following properties: {displaystyle varnothing in {matematico {S}}} Per tutti {stile di visualizzazione A,Bin {matematico {S}},} noi abbiamo {displaystyle Acap Bin {matematico {S}}} (closed under pairwise intersections) Per tutti {stile di visualizzazione A,Bin {matematico {S}},} there exist disjoint sets {stile di visualizzazione K_{io}in {matematico {S}},i=1,2,ldots ,n,} tale che {displaystyle Asetminus B=bigcup _{io=1}^{n}K_{io}} (relative complements can be written as finite disjoint unions).

The first property can be replaced with {stile di visualizzazione {matematico {S}}neq varnothing } da {displaystyle Ain {matematico {S}}implies Asetminus A=varnothing in {matematico {S}}.} With the same notation, we call a family {stile di visualizzazione {matematico {R}}} of subsets of {stile di visualizzazione Omega } a ring of sets if it has the following properties: {displaystyle varnothing in {matematico {R}}} Per tutti {stile di visualizzazione A,Bin {matematico {R}},} noi abbiamo {displaystyle Acup Bin {matematico {R}}} (closed under pairwise unions) Per tutti {stile di visualizzazione A,Bin {matematico {R}},} noi abbiamo {displaystyle Asetminus Bin {matematico {R}}} (closed under relative complements).

così, any ring on {stile di visualizzazione Omega } is also a semi-ring.

A volte, the following constraint is added in the measure theory context: {stile di visualizzazione Omega } is the disjoint union of a countable family of sets in {stile di visualizzazione {matematico {S}}.} A field of sets (rispettivamente, a semi-field) is a ring (rispettivamente, a semi-ring) that also contains {stile di visualizzazione Omega } as one of its elements.

Properties Arbitrary (possibly uncountable) intersections of rings on {stile di visualizzazione Omega } are still rings on {stile di visualizzazione Omega .} Se {stile di visualizzazione A} is a non-empty subset of the powerset {stile di visualizzazione {matematico {P}}(Omega )} di {stile di visualizzazione Omega ,} then we define the ring generated by {stile di visualizzazione A} (noted {stile di visualizzazione R(UN)} ) as the intersection of all rings containing {displaystyle A.} It is straightforward to see that the ring generated by {stile di visualizzazione A} is the smallest ring containing {displaystyle A.} For a semi-ring {stile di visualizzazione S,} the set of all finite unions of sets in {stile di visualizzazione S} is the ring generated by {stile di visualizzazione S:} {stile di visualizzazione R(S)= sinistra{UN:A=bigcup _{io=1}^{n}UN_{io},UN_{io}in Sright}} (One can show that {stile di visualizzazione R(S)} is equal to the set of all finite disjoint unions of sets in {stile di visualizzazione S} ). A content {displaystyle lui } defined on a semi-ring {stile di visualizzazione S} can be extended on the ring generated by {displaystyle S.} Such an extension is unique. The extended content can be written: {displaystyle lui (UN)=somma _{io=1}^{n}in (UN_{io})} per {displaystyle A=bigcup _{io=1}^{n}UN_{io},} with the {stile di visualizzazione A_{io}a s} disjoint.

Inoltre, it can be proved that {displaystyle lui } is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on {stile di visualizzazione R(S)} that extends the pre-measure on {stile di visualizzazione 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 {stile di visualizzazione S} (for example Stieltjes measures), which can then be extended to a pre-measure on {stile di visualizzazione 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). In realtà, 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 {stile di visualizzazione {matematico {P}}(mathbb {R} )} defined by the set of all half-open intervals {stile di visualizzazione [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.

Enunciato del teorema Let {stile di visualizzazione R} be a ring of sets on {stile di visualizzazione X} e lascia {displaystyle lui :Rto [0,+infty ]} be a pre-measure on {stile di visualizzazione R,} meaning that for all sets {displaystyle Ain R} for which there exists a countable decomposition {displaystyle A=bigcup _{io=1}^{infty }UN_{io}} in disjoint sets {stile di visualizzazione A_{1},UN_{2},ldots in R,} noi abbiamo {displaystyle lui (UN)=somma _{io=1}^{infty }in (UN_{io}).} Permettere {displaystyle sigma (R)} be the {displaystyle sigma } -algebra generated by {stile di visualizzazione R.} The pre-measure condition is a necessary condition for {displaystyle lui } to be the restriction to {stile di visualizzazione R} of a measure on {displaystyle sigma (R).} The Carathéodory's extension theorem states that it is also sufficient,[3] questo è, there exists a measure {stile di visualizzazione mu ^{primo }:sigma (R)a [0,+infty ]} tale che {stile di visualizzazione mu ^{primo }} is an extension of {displaystyle lui ;} questo è, {stile di visualizzazione mu ^{primo }{big vert }_{R}=mu .} Inoltre, Se {displaystyle lui } è {displaystyle sigma } -finite then the extension {stile di visualizzazione mu ^{primo }} è unico (and also {displaystyle sigma } -finito).[4] Proof sketch First extend {displaystyle lui } to an outer measure {stile di visualizzazione mu ^{*}} on the power set {stile di visualizzazione 2 ^{X}} di {stile di visualizzazione X} di {stile di visualizzazione mu ^{*}(T)=inf a sinistra{somma _{n}mu left(S_{n}Giusto):Tsubseteq cup _{n}S_{n}{testo{ insieme a }}S_{1},S_{2},ldots in Rright}} and then restrict it to the set {stile di visualizzazione {matematico {B}}} di {stile di visualizzazione mu ^{*}} -measurable sets (questo è, Carathéodory-measurable sets), which is the set of all {displaystyle Msubseteq X} tale che {stile di visualizzazione mu ^{*}(S)=mu ^{*}(Scap M)+in ^{*}(Scap M^{matematica {c} })} per ogni {displaystyle Ssubseteq X.} It is a {displaystyle sigma } -algebra, e {stile di visualizzazione mu ^{*}} è {displaystyle sigma } -additive on it, by the Caratheodory lemma.

It remains to check that {stile di visualizzazione {matematico {B}}} contains {stile di visualizzazione R.} Questo è, to verify that every set in {stile di visualizzazione R} è {stile di visualizzazione mu ^{*}} -measurable. This is done by basic measure theory techniques of dividing and adding up sets.

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

Now we could concretely prove {displaystyle nu =mu ^{*}} Su {displaystyle sigma (R)} by using the Borel hierarchy of {stile di visualizzazione R,} e da allora {displaystyle nu =mu ^{*}} at the base level, we can use well-ordered induction to reach the level of {stile di visualizzazione 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 [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 {displaystyle mathbb {Q} } della forma {stile di visualizzazione [un,b)} , dove {stile di visualizzazione a,sono matematicabb {Q} } .

Permettere {stile di visualizzazione X} be {displaystyle mathbb {Q} berretto [0,1)} e lascia {displaystyle Sigma _{0}} be the algebra of all finite unions of rational closed-open intervals contained in {displaystyle mathbb {Q} berretto [0,1)} . It is easy to prove that {displaystyle Sigma _{0}} è, infatti, an algebra. It is also easy to see that the cardinal of every non-empty set in {displaystyle Sigma _{0}} è {stile di visualizzazione aleph _{0}} .

Permettere {displaystyle lui _{0}} be the counting set function ( {stile di visualizzazione #} ) defined in {displaystyle Sigma _{0}} . È chiaro che {displaystyle lui _{0}} is finitely additive and {displaystyle sigma } -additive in {displaystyle Sigma _{0}} . Since every non-empty set in {displaystyle Sigma _{0}} is infinite, poi, for every non-empty set {displaystyle Ain Sigma _{0}} , {displaystyle lui _{0}(UN)=+infty } Adesso, permettere {stile di visualizzazione Sigma } be the {displaystyle sigma } -algebra generated by {displaystyle Sigma _{0}} . It is easy to see that {stile di visualizzazione Sigma } is the Borel {displaystyle sigma } -algebra of subsets of {stile di visualizzazione X} , and both {stile di visualizzazione #} e {stile di visualizzazione 2#} are measures defined on {stile di visualizzazione Sigma } and both are extensions of {displaystyle 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; Per esempio: 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. Tuttavia, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) chi l'ha dimostrato 1924." Paul Loya (pagina 33). ^ Klenke, Achim (2014). Probability Theory. Universitext. 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, Roberto B. (1999). Probability and Measure Theory (2nd ed.). Stampa accademica. p. 19. ISBN 0-12-065202-1.

This article incorporates material from Hahn–Kolmogorov theorem on PlanetMath, che è concesso in licenza in base alla licenza Creative Commons Attribution/Share-Alike.

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