Maharam’s problem.

*(English)*Zbl 1185.28002This 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” (BirkhĂ¤user, 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.

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” (BirkhĂ¤user, 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.

Reviewer: D. R. Bell (Jacksonville)

##### MSC:

28A12 | Contents, measures, outer measures, capacities |