Between Maharam's and von Neumann's problems

Abstract

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 σ-ideals with the Fubini property are (meager,meager) and (null,null).