Between Maharam’s and von Neumann’s problems

In the context of definable algebras Maharam’s and von Neumann’s problems essentially coincide. Consequently, random forcing is the only definable ccc forcing adding a single real that does not make the ground model reals null, and the only pairs of definable ccc $\sigma$-ideals with the Fubini property are $(\mathtt{meager},\mathtt{meager})$ and $(\mathtt{null},\mathtt{null})$.

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Published 1 January 2004