Functional Analysis
Maharam's problem
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
Talagrand, Michel 1

1 Équipe d'analyse de l'institut mathématique, université Paris VI, boîte 186, 4, place Jussieu, 75230 Paris cedex 05, France
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Talagrand, Michel. Maharam's problem. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 501-503. doi : 10.1016/j.crma.2006.01.026. http://www.numdam.org/articles/10.1016/j.crma.2006.01.026/

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

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

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[4] J.W. Roberts, Maharam's problem, in: Kranz, Labuda (Eds.), Procedings of the Orlicz Memorial Conference, 1991, unpublished

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