Comptes Rendus
no. 7
Functional Analysis
Maharam's problem
Gilles Pisier
Michel Talagrand1
1Équipe 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 
Michel Talagrand. Maharam's problem. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 501-503. doi: 10.1016/j.crma.2006.01.026
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[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[4] J.W. Roberts, Maharam's problem, in: Kranz, Labuda (Eds.), Procedings of the Orlicz Memorial Conference, 1991, unpublished

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