Choquet's continuation

The continuation of Choquet is a mathematical theorem , which is located in the transition field between the field of measure theory and the field of functional analysis and which goes back to the mathematician Gustave Choquet . He shows that for a Hausdorff space an internally regular locally finite measure on the associated Borel σ-algebra is already unambiguously determined by the associated real set function on the set system of the compact subsets , provided that this set function alone already has certain simple (and obvious) conditions are sufficient. The Choquetian continuation clause is closely linked to the representation clause of Riesz-Markov-Kakutani and its meaning lies not least in the fact that the representation clause can be traced back to it.

Formulation of the sentence

The sentence can be formulated as follows:

A Hausdorff space is given , provided with Borel's σ-algebra and the set system of compact subsets of ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle {\mathcal {K}}(X)\subseteq {\mathcal {B}}(X)}$ ${\displaystyle X}$ .

A set function is also given

${\displaystyle {\mu }_{0}\colon {\mathcal {K}}(X)\rightarrow \mathbb {R} ^{\geq 0}}$ ,

which may meet the following conditions:

(R_1) For with always applies ${\displaystyle K,L\in {\mathcal {K}}(X)}$ ${\displaystyle K\subseteq L}$

${\displaystyle {\mu }_{0}(K)\leq {\mu }_{0}(L)}$ .

(R_2) For always applies ${\displaystyle K,L\in {\mathcal {K}}(X)}$

${\displaystyle {\mu }_{0}(K\cup L)\leq {\mu }_{0}(K)+{\mu }_{0}(L)}$ .

(R_3) For with always applies ${\displaystyle K,L\in {\mathcal {K}}(X)}$ ${\displaystyle K\cap L=\emptyset }$

${\displaystyle {\mu }_{0}{(K\cup L)}={\mu }_{0}(K)+{\mu }_{0}(L)}$.

(R_4) To and there is always an open environment in such a way that applies to everyone with : ${\displaystyle K\in {\mathcal {K}}(X)}$ ${\displaystyle \epsilon >0}$ ${\displaystyle U}$ ${\displaystyle K}$ ${\displaystyle L\in {\mathcal {K}}(X)}$ ${\displaystyle L\subseteq U}$

${\displaystyle {\mu }_{0}(L)\leq {\mu }_{0}(K)+\epsilon }$ .

Under these conditions there can be exactly one way to an internally regular locally finite measure ${\displaystyle {\mu }_{0}}$

${\displaystyle \mu \colon {\mathcal {B}}(X)\rightarrow \mathbb {R} ^{\geq 0}\cup \{\infty \}}$

to be continued .

So it is

${\displaystyle {\mu }_{0}={\mu }|_{{\mathcal {K}}(X)}}$

and one has for everyone ${\displaystyle A\in {\mathcal {B}}(X)}$

${\displaystyle \mu (A)=\sup\{{\mu }_{0}(K):K\in {\mathcal {K}}(X)\;,\;K\subseteq A\}}$ .

Explanations and Notes

• Ehrhard Behrends and Jürgen Elstrodt (and just as many other authors) refer to the measures mentioned in the Choquetian continuation theorem on Borel σ-algebras of Hausdorff spaces as Radon measures . ${\displaystyle \mu }$
• So the sequel set includes the statement that and the conditions defined on the system of compact subsets of a Hausdorff space the Radon measures, (R_1) (R_2) (R_3) (R_4) sufficient amount of functions each reversible clearly address .
• In his presentation of the continuation, Elstrodt shows - following the 1968 work On the generation of tight measures by the Polish mathematician Jan Kisyński - that the condition (R_4) can be replaced by the so-called tightness condition (S) , which states the following:

(S) For with always applies ${\displaystyle K,L\in {\mathcal {K}}(X)}$ ${\displaystyle K\subseteq L}$

${\displaystyle {\mu }_{0}(L)-{\mu }_{0}(K)=\sup\{{\mu }_{0}(C):C\in {\mathcal {K}}(X)\;,\;C\subseteq {L\setminus K}\}}$ .

• As Elstrodt notes, there are different variants of the continuation clause in the specialist literature. The lively research activity that has arisen has shown that the punishment condition is of essential importance with regard to the question of continuability.

Sources and background literature

• Ehrhard Behrends: Measure and integration theory (=  university text ). Springer-Verlag , Berlin, Heidelberg, New York, London, Paris, Tokyo 1987, ISBN 3-540-17850-3 ( MR1028059 ).
• Jürgen Elstrodt: Measure and integration theory (=  Springer textbook - basic knowledge of mathematics ). 7th, corrected and updated edition. Springer-Verlag , Heidelberg, Dordrecht, London, New York 2011, ISBN 978-3-642-17904-4 , doi : 10.1007 / 978-3-642-17905-1 .
• J. Kisyński : On the generation of tight measures . In: Studia Mathematica . tape 30 , 1968, pp. 141-151 ( [1] ).
• Dirk Werner : Functional Analysis . 6th, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .

Individual evidence

1. ↑ Ehrhard Behrends: Measure and integration theory. 1987, p. 205 ff
2. ↑ Jürgen Elstrodt: Measure and integration theory. 2011, p. 328 ff
3. ↑ Dirk Werner: Functional Analysis. 2007, pp. 89-90
4. ↑ Behrends, op. Cit., Pp. 206-207
5. ↑ Elstrodt, op. Cit., Pp. 331-332
6. ↑ Behrends, op.cit., P. 196
7. ↑ ^{a b} Elstrodt, op.cit., P. 313
8. ↑ Elstrodt, op.cit., P. 331 ff