Comptes Rendus
no. 7
Functional Analysis
Maharam's problem

Presented by: Gilles Pisier

Michel Talagrand1
1 Équipe d'analyse de l'institut mathématique, université Paris VI, boîte 186, 4, place Jussieu, 75230 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 501-503.

We solve Maharam's problem [D. Maharam, An algebraic characterization of measure algebras, Ann. Math. 48 (1947) 154–167. [3]], also known as the Control Measure Problem. We construct a non-zero exhaustive submeasure on the algebra of clopen sets of the Cantor set that is not absolutely continuous with respect to a measure.

Nous construisons une sous-mesure exhaustive non nulle sur l'algèbre des ouverts-fermés de l'ensemble de Cantor qui n'est absolument continue par rapport à aucune mesure. Ceci résout un problème posé par D. Maharam [D. Maharam, An algebraic characterization of measure algebras, Ann. Math. 48 (1947) 154–167. [3]], en 1947, aussi connu sous le nom de problème des mesures de contrôle.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.01.026

Michel Talagrand 1

1 Équipe d'analyse de l'institut mathématique, université Paris VI, boîte 186, 4, place Jussieu, 75230 Paris cedex 05, France 
@article{CRMATH_2006__342_7_501_0,
     author = {Michel Talagrand},
     title = {Maharam's problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {501--503},
     publisher = {Elsevier},
     volume = {342},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.026},
     language = {en},
}
TY  - JOUR
AU  - Michel Talagrand
TI  - Maharam's problem
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 501
EP  - 503
VL  - 342
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2006.01.026
LA  - en
ID  - CRMATH_2006__342_7_501_0
ER  - 
%0 Journal Article
%A Michel Talagrand
%T Maharam's problem
%J Comptes Rendus. Mathématique
%D 2006
%P 501-503
%V 342
%N 7
%I Elsevier
%R 10.1016/j.crma.2006.01.026
%G en
%F CRMATH_2006__342_7_501_0
Michel Talagrand. Maharam's problem. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 501-503. doi : 10.1016/j.crma.2006.01.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.026/

[1] I. Farah Example of ɛ-exhaustive pathological measures, Fund. Math., Volume 181 (2004), pp. 257-272

[2] D.H. Fremlin Measure Theory, Vol. 3: Measure Algebras, Torres Fremlin, 2004

[3] D. Maharam An algebraic characterization of measure algebras, Ann. Math., Volume 48 (1947), pp. 154-167

[4] J.W. Roberts, Maharam's problem, in: Kranz, Labuda (Eds.), Procedings of the Orlicz Memorial Conference, 1991, unpublished

Cited by Sources:

Comments - Policy