Dimension extension set by Carathéodory

The Carathéodory's extension theorem is a phrase from the mathematical branch of measure theory . This theorem serves to extend premeasures , which are defined on set rings , to measures on σ-algebras . With this method, which goes back to Constantin Carathéodory , the Lebesgue measure in particular can be traced back to the determination of the length of intervals.

Formulation of the sentence

Let it be a premeasure on a set of sets from a basic set . Then there is a comprehensive σ-algebra on and an extension from to a measure on such that is a complete measure space . ${\displaystyle \mu }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle X}$ ${\displaystyle {\tilde {\mu }}}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {A}},{\tilde {\mu }})}$

construction

By means of the premeasure given on the ring, one defines an outer measure that is defined on the entire power set and from that, by means of a suitable restriction, a measure on a σ-algebra. This construction will now be described in detail and applied in parallel to the construction of the Lebesgue measure. ${\displaystyle {\mathcal {P}}(X)}$

Dimensions on rings

A set ring contains the empty set and is closed with regard to finite unions and the formation of difference sets. A premeasure on such a set ring is a function with and , if pairwise disjoint sets are off, their union is again in . The standard example is the set of all finite unions of half-open intervals in , where always be. Such associations can always as a disjoint union is written such intervals, and the fixing , wherein the length is such an interval, defined on a premeasure , known Lebesgue premeasure . ${\displaystyle \mu \colon {\mathcal {R}}\rightarrow [0,\infty ]}$ ${\displaystyle \mu (\emptyset )=0}$ ${\displaystyle \textstyle \mu (\bigcup _{n}A_{n})=\sum _{n}\mu (A_{n})}$ ${\displaystyle A_{1},A_{2},\ldots }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}^{1}}$ ${\displaystyle I=[a,b)}$ ${\displaystyle X=\mathbb {R} }$ ${\displaystyle a\leq b}$ ${\displaystyle I_{1}\cup \ldots \cup I_{k}}$ ${\displaystyle \mu (I_{1}\cup \ldots \cup I_{k})=\ell (I_{1})+\ldots +\ell (I_{k})}$ ${\displaystyle \ell ([a,b))=b-a}$ ${\displaystyle {\mathcal {R}}^{1}}$

This is easily generalized to dimensions if one looks at the set ring of all finite unions of n-dimensional intervals ( cuboids ) , where always be. Here, too, one can restrict oneself to disjoint unions and in such a case ${\displaystyle n}$ ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {R}}^{n}}$ ${\displaystyle Q=[a_{1},b_{1})\times \ldots \times [a_{n},b_{n})}$ ${\displaystyle a_{i}\leq b_{i}}$

${\displaystyle \mu (Q_{1}\cup \ldots \cup Q_{k})=vol(Q_{1})+\ldots +vol(Q_{k})}$

define, where the usual elementary geometric volume of a cuboid is. This example is also called the n-dimensional Lebesgue premeasure. ${\displaystyle vol([a_{1},b_{1})\times \ldots \times [a_{n},b_{n}))=(b_{1}-a_{1})\cdot \ldots \cdot (b_{n}-a_{n})}$

Construction of the outer dimension

Let a content be given on a set ring of sets from a basic set . For every subset let ${\displaystyle \mu }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle X}$ ${\displaystyle Y\subset X}$

${\displaystyle \mu ^{*}(Y):=\inf \left\{\sum _{k=1}^{\infty }\mu (A_{k});\,A_{1},A_{2},\ldots \in {\mathcal {R}},Y\subset \bigcup _{k=1}^{\infty }A_{k}\right\}}$

whereby . Then there is an external measure . You can show that ${\displaystyle \inf(\emptyset ):=\infty }$ ${\displaystyle \mu ^{*}}$ ${\displaystyle {\mathcal {P}}(X)}$

1. ${\displaystyle \mu ^{*}(A)=\mu (A)}$
2. ${\displaystyle \mu ^{*}(Q)=\mu ^{*}(Q\cap A)+\mu ^{*}(Q\cap (X\setminus A))}$

for everyone and . The first property says that the given measure continues, the second that every amount of the base space is divided into two parts by every amount of the given ring, which behave additively. ${\displaystyle Q\subset X}$ ${\displaystyle A\in {\mathcal {R}}}$ ${\displaystyle \mu ^{*}}$ ${\displaystyle \mu ^{*}}$

Transition to measurable quantities

The core of Carathéodory's construction is the definition of

${\displaystyle {\mathcal {A}}:=\{A\subset X;\mu ^{*}(Q)=\mu ^{*}(Q\cap A)+\mu ^{*}(Q\cap (X\setminus A)){\text{ für alle }}Q\subset X\}}$,

proof that this defines a σ-algebra, the so-called σ-algebra of the Carathéodory measurable sets , and that the restriction is a measure. Because of the above second property of the external dimension and because of the first is a continuation of . Finally one shows that every set with external measure contains 0, from which the completeness of the measure space results. ${\displaystyle {\tilde {\mu }}:=\mu ^{*}|_{\mathcal {A}}}$ ${\displaystyle {\mathcal {R}}\subset {\mathcal {A}}}$ ${\displaystyle {\tilde {\mu }}}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {A}},{\tilde {\mu }})}$

If one applies this construction to our example of the Lebesgue premeasure, one obtains the Lebesgue measure on the Lebesgue σ-algebra. In this case one can show that Lebesgue's σ-algebra is genuinely larger than the σ-algebra generated by , which coincides with Borel's σ-algebra . However, the difference is not too great, because every set of Lebesgue's σ-algebra differs only by a -zero set from a Borel set, i.e. Lebesgue's σ-algebra is the completion of Borel. ${\displaystyle {\mathcal {R}}^{n}}$ ${\displaystyle \mu ^{*}}$

Remarks

Uniqueness

As a consequence of the above theorem one obtains that every premeasure on a ring can be continued to a measure on the σ-algebra generated by the ring. One obtains a statement of uniqueness from the uniqueness theorem if one additionally presupposes that finite pre-measure can be written as a countable union of ring sets, i.e. the pre-measure is -finite . ${\displaystyle X}$ ${\displaystyle \sigma }$

Size of the continuation

It can be shown that if the external measure used for the construction is a metric external measure , then the σ-algebra of the measurable sets contains the Borel σ-algebra . This is a further explanation for the fact that Lebesgue σ-algebra is really larger than Borel’s σ-algebra.

Half rings

Instead of quantity rings, one can also start from the more general term half-ring . A measure or premeasure on a half-ring is defined as on rings, that is, it is a set function , so that and , if pairwise disjoint sets are off, their union is again in . ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mu \colon {\mathcal {H}}\rightarrow [0,\infty ]}$ ${\displaystyle \mu (\emptyset )=0}$ ${\displaystyle \textstyle \mu (\bigcup _{n}A_{n})=\sum _{n}\mu (A_{n})}$ ${\displaystyle A_{1},A_{2},\ldots }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {H}}}$

In order to get a measure extension in this situation, one first forms the ring generated by , which is equal to the set of all finite, disjoint unions of sets from . If there is such a disjoint union, a measure on the set ring is explained by the fixing . The construction described above can then be applied to this. ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {R}}_{\mathcal {H}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle C_{1}\cup \ldots \cup C_{k}}$ ${\displaystyle {\hat {\mu }}(C_{1}\cup \ldots \cup C_{k}):=\mu (C_{1})+\ldots +\mu (C_{k})}$ ${\displaystyle {\hat {\mu }}}$ ${\displaystyle {\mathcal {R}}_{\mathcal {H}}}$

The standard example is the half-ring of all half-open n-dimensional intervals (cuboid) ${\displaystyle {\mathcal {H}}^{n}}$

${\displaystyle [a,b):=\{X=(x_{1},\ldots ,x_{n});\,a_{i}\leq x_{i}<b_{i}{\text{ für alle }}i=1,\ldots ,n\}}$

with and the stated degree of elementary geometric content. The construction presented here leads directly from the definition of the cuboid volume as the product of the side lengths to the Lebesgue dimension. It can be generalized directly to general product dimensions . ${\displaystyle a=(a_{1},\ldots ,a_{n}),b=(b_{1},\ldots ,b_{n})\in \mathbb {R} ^{n}}$

Generalizations

More generally, it can be shown that if a half-ring and an additive, -subadditive and -endliche set function , then there exists a unique continuation of on , which is a measure and conforms on each amount of the half-ring with the set function. This formulation contains the above as a special case. ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \mu }$ ${\displaystyle \sigma ({\mathcal {H}})}$

Individual evidence

1. ↑ Ernst Henze : Introduction to the measurement theory ( BI university pocket books. Vol. 505). Bibliographisches Institut , Mannheim et al. 1971, ISBN 3-411-00505-X , chap. 2.
2. ↑ Heinz Bauer : Probability Theory. 4th, completely revised and redesigned edition. de Gruyter, Berlin 1991, ISBN 3-11-012191-3 , chap. I, § 4.
3. ↑ Heinz Bauer: Probability Theory. 4th, completely revised and redesigned edition. de Gruyter, Berlin 1991, ISBN 3-11-012191-3 , chap. I, sentence 5.2.
4. ↑ Heinz Bauer: Probability Theory. 4th, completely revised and redesigned edition. de Gruyter, Berlin 1991, ISBN 3-11-012191-3 , chap. I, sentence 5.4.
5. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , sentence 2.1.9.
6. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , sentence 1.5.2
7. ↑ Ernst Henze: Introduction to the measurement theory ( BI university pocket books. Vol. 505). Bibliographisches Institut, Mannheim et al. 1971, ISBN 3-411-00505-X , chap. 2.
8. ↑ Ernst Henze: Introduction to the measurement theory ( BI university pocket books. Vol. 505). Bibliographisches Institut, Mannheim et al. 1971, ISBN 3-411-00505-X , chap. 1.5, sentence 6.
9. ↑ Ernst Henze: Introduction to the measurement theory ( BI university pocket books. Vol. 505). Bibliographisches Institut, Mannheim et al. 1971, ISBN 3-411-00505-X , chap. 2.3, sentence 2.