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Maharam’s problem. (English) Zbl 1185.28002
This paper settles in the negative several long standing conjectures in measure theory. The author constructs (1) a non-zero exhaustive submeasure $$\nu$$ on the algebra $$B$$ of clopen subsets of the Cantor set that is not absolutely continuous with respect to a measure on $$B$$. Furthermore, no non-zero measure on $$B$$ is absolutely continuous with respect to $$\nu$$. This construction gives a negative answer to a problem of D. Maharam [Ann. Math. (2) 48, 154–167 (1947; Zbl 0029.20401)].
The author’s work yields other important results. He proves (2) there exists a $$\sigma$$-complete algebra that satisfies the countable chain condition and is weakly distributive but is not a measure algebra. This gives a counterexample to a conjecture of von Neumann of 1937 [cf. R. D. Mauldin (ed.), “The Scottish book. Mathematics from the Scottish Cafe” (Birkhäuser, Boston–Basel–Stuttgart) (1981; Zbl 0485.01013)]. (3) There exists an exhaustive measure that does not have a control measure, a negative solution to the Control Measure Problem.
The paper contains a good presentation of crucial background material due to J. W. Roberts and I. Farah, making it quite self-contained and readable.

##### MSC:
 28A12 Contents, measures, outer measures, capacities
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