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Maharam’s problem. (English) Zbl 1099.28004

It is a long standing problem in measure theory whether any exhaustive submeasure on an algebra is continuous with respect to a positive real-valued measure. This problem is known as Maharam’s control measure problem and was posed (in an equivalent version) by D. Maharam [Ann. Math. (2) 48, 154–167 (1947; Zbl 0029.20401)]. The author gives a negative answer to this problem constructing a suitable submeasure on the algebra of clopen sets of the Cantor set.
Reviewer: Hans Weber (Udine)

MSC:

28A12 Contents, measures, outer measures, capacities

Citations:

Zbl 0029.20401
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References:

[1] Farah, I., Example of \(ϵ\)-exhaustive pathological measures, Fund. Math., 181, 257-272 (2004) · Zbl 1069.28002
[2] Fremlin, D. H., Measure Theory, Vol. 3: Measure Algebras (2004), Torres Fremlin · Zbl 1165.28002
[3] Maharam, D., An algebraic characterization of measure algebras, Ann. Math., 48, 154-167 (1947) · Zbl 0029.20401
[4] J.W. Roberts, Maharam’s problem, in: Kranz, Labuda (Eds.), Procedings of the Orlicz Memorial Conference, 1991, unpublished; J.W. Roberts, Maharam’s problem, in: Kranz, Labuda (Eds.), Procedings of the Orlicz Memorial Conference, 1991, unpublished · Zbl 0744.28002
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