# Erweiterungssatz von Carathéodory

In measure theory, Erweiterungssatz von 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. Folglich, 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, zum Beispiel, 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] Inhalt 1 Introductory statement 1.1 Kommentare 2 Semi-ring and ring 2.1 Definitionen 2.2 Eigenschaften 2.3 Motivation 2.4 Beispiel 3 Aussage des Theorems 3.1 Beweisskizze 4 Examples of non-uniqueness of extension 4.1 Via the counting measure 4.2 Via rationals 4.3 Via Fubini's theorem 5 Siehe auch 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. Ein kürzer, simpler statement is as follows. In dieser Form, it is often called the Hahn–Kolmogorov theorem.

Lassen {Anzeigestil Sigma _{0}} be an algebra of subsets of a set {displaystyle X.} Consider a set function {displaystyle ihn _{0}:Sigma _{0}zu [0,unendlich ]} which is finitely additive, bedeutet, dass {displaystyle ihn _{0}links(große Tasse _{n=1}^{N}EIN_{n}Rechts)= Summe _{n=1}^{N}in _{0}(EIN_{n})} for any positive integer {Anzeigestil N} und {Anzeigestil A_{1},EIN_{2},Punkte ,EIN_{N}} disjoint sets in {Anzeigestil Sigma _{0}.} Assume that this function satisfies the stronger sigma additivity assumption {displaystyle ihn _{0}links(große Tasse _{n=1}^{unendlich }EIN_{n}Rechts)= Summe _{n=1}^{unendlich }in _{0}(EIN_{n})} for any disjoint family {Anzeigestil {EIN_{n}:nin mathbb {N} }} von Elementen von {Anzeigestil Sigma _{0}} so dass {displaystyle cup _{n=1}^{unendlich }EIN_{n}in Sigma _{0}.} (Functions {displaystyle ihn _{0}} obeying these two properties are known as pre-measures.) Dann, {displaystyle ihn _{0}} extends to a measure defined on the {Display-Sigma } -algebra {displaystyle Sigma } generated by {Anzeigestil Sigma _{0}} ; das ist, there exists a measure {zeige ihn an :Sigma to [0,unendlich ]} such that its restriction to {Anzeigestil Sigma _{0}} coincides with {displaystyle ihn _{0}.} Wenn {displaystyle ihn _{0}} ist {Display-Sigma } -endlich, 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 ihn _{0}} from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (wenn {displaystyle ihn _{0}} ist {Display-Sigma } -endlich), 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 {Anzeigestil Omega ,} we call a family {Anzeigestil {mathematisch {S}}} of subsets of {Anzeigestil Omega } a semi-ring of sets if it has the following properties: {displaystyle varnothing in {mathematisch {S}}} Für alle {Anzeigestil A,Bin {mathematisch {S}},} wir haben {displaystyle Acap Bin {mathematisch {S}}} (closed under pairwise intersections) Für alle {Anzeigestil A,Bin {mathematisch {S}},} there exist disjoint sets {Anzeigestil K_{ich}in {mathematisch {S}},i=1,2,ldots ,n,} so dass {displaystyle Asetminus B=bigcup _{i=1}^{n}K_{ich}} (relative complements can be written as finite disjoint unions).

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

Daher, any ring on {Anzeigestil Omega } is also a semi-ring.

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

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

Zusätzlich, it can be proved that {zeige ihn an } is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on {Anzeigestil R(S)} that extends the pre-measure on {Anzeigestil 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 {Anzeigestil S} (for example Stieltjes measures), which can then be extended to a pre-measure on {Anzeigestil 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). Eigentlich, 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 {Anzeigestil {mathematisch {P}}(mathbb {R} )} defined by the set of all half-open intervals {Anzeigestil [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.

Aussage des Theorems Let {Anzeigestil R} be a ring of sets on {Anzeigestil X} und lass {zeige ihn an :Rto [0,+unendlich ]} be a pre-measure on {Anzeigestil R,} meaning that for all sets {displaystyle Ain R} for which there exists a countable decomposition {displaystyle A=bigcup _{i=1}^{unendlich }EIN_{ich}} in disjoint sets {Anzeigestil A_{1},EIN_{2},ldots in R,} wir haben {zeige ihn an (EIN)= Summe _{i=1}^{unendlich }in (EIN_{ich}).} Lassen {Display-Sigma (R)} be the {Display-Sigma } -algebra generated by {Anzeigestil R.} The pre-measure condition is a necessary condition for {zeige ihn an } to be the restriction to {Anzeigestil R} of a measure on {Display-Sigma (R).} The Carathéodory's extension theorem states that it is also sufficient,[3] das ist, there exists a measure {displaystyle mu ^{prim }:Sigma (R)zu [0,+unendlich ]} so dass {displaystyle mu ^{prim }} is an extension of {zeige ihn an ;} das ist, {displaystyle mu ^{prim }{big vert }_{R}= ein .} Darüber hinaus, wenn {zeige ihn an } ist {Display-Sigma } -finite then the extension {displaystyle mu ^{prim }} ist einzigartig (and also {Display-Sigma } -endlich).[4] Proof sketch First extend {zeige ihn an } to an outer measure {displaystyle mu ^{*}} on the power set {Anzeigestil 2 ^{X}} von {Anzeigestil X} durch {displaystyle mu ^{*}(T)=inf left{Summe _{n}Ich bin gegangen(S_{n}Rechts):Tsubseteq cup _{n}S_{n}{Text{ mit }}S_{1},S_{2},ldots in Rright}} and then restrict it to the set {Anzeigestil {mathematisch {B}}} von {displaystyle mu ^{*}} -measurable sets (das ist, Carathéodory-measurable sets), which is the set of all {displaystyle Msubseteq X} so dass {displaystyle mu ^{*}(S)=mu ^{*}(Scap M)+mu ^{*}(Scap M^{Mathrm {c} })} für jeden {displaystyle Ssubseteq X.} It is a {Display-Sigma } -algebra, und {displaystyle mu ^{*}} ist {Display-Sigma } -additive on it, by the Caratheodory lemma.

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

For uniqueness, take any other extension {Anzeigestil Nr } so it remains to show that {displaystyle nu =mu ^{*}.} Durch {Display-Sigma } -additivity, uniqueness can be reduced to the case where {zeige ihn an (X)} ist endlich, which will now be assumed.

Now we could concretely prove {displaystyle nu =mu ^{*}} an {Display-Sigma (R)} by using the Borel hierarchy of {Anzeigestil R,} und da {displaystyle nu =mu ^{*}} at the base level, we can use well-ordered induction to reach the level of {displaystyle omega _{1},} the level of {Display-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 {Anzeigestil mathbb {Q} } des Formulars {Anzeigestil [a,b)} , wo {Anzeigestil a,bin mathbb {Q} } .

Lassen {Anzeigestil X} be {Anzeigestil mathbb {Q} cap [0,1)} und lass {Anzeigestil Sigma _{0}} be the algebra of all finite unions of rational closed-open intervals contained in {Anzeigestil mathbb {Q} cap [0,1)} . It is easy to prove that {Anzeigestil Sigma _{0}} ist, tatsächlich, an algebra. It is also easy to see that the cardinal of every non-empty set in {Anzeigestil Sigma _{0}} ist {Anzeigestil Aleph _{0}} .

Lassen {displaystyle ihn _{0}} be the counting set function ( {Anzeigestil #} ) defined in {Anzeigestil Sigma _{0}} . Es ist klar, dass {displaystyle ihn _{0}} is finitely additive and {Display-Sigma } -additive in {Anzeigestil Sigma _{0}} . Since every non-empty set in {Anzeigestil Sigma _{0}} ist unendlich, dann, for every non-empty set {displaystyle Ain Sigma _{0}} , {displaystyle ihn _{0}(EIN)=+infty } Jetzt, Lassen {displaystyle Sigma } be the {Display-Sigma } -algebra generated by {Anzeigestil Sigma _{0}} . It is easy to see that {displaystyle Sigma } is the Borel {Display-Sigma } -algebra of subsets of {Anzeigestil X} , and both {Anzeigestil #} und {Anzeigestil 2#} are measures defined on {displaystyle Sigma } and both are extensions of {displaystyle ihn _{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(EIN)card(B). This has a very large number of different extensions to a measure; zum Beispiel: 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 {Anzeigestil 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. Jedoch, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) der es bewiesen hat 1924." Paul Loya (Seite 33). ^ Klenke, Achim (2014). Probability Theory. Universitätstext. p. Satz 1.53. doi:10.1007/978-1-4471-5361-0. ISBN 978-1-4471-5360-3. ^ Vaillant, Noel. "Caratheodory's Extension" (Pdf). Probability.net. Satz 4. ^ Ash, Robert B. (1999). Probability and Measure Theory (2und Aufl.). Akademische Presse. p. 19. ISBN 0-12-065202-1.