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 -ideals with the Fubini property are (meager,meager) and (null,null). In Scottish Book, von Neumann asked whether every ccc, weakly distributive complete Boolean algebra carries a strictly positive probability measure. Von Neumann’s problem naturally splits into two: (a) whether all such algebras carry a strictly positive continuous submeasure, and (b) whether every algebra that carries a strictly positive continuous submeasure carries a strictly positive measure. The latter problem is known under the names of Maharam’s Problem and Control Measure Problem (see [16], [9], [5, §393]). While von Neumann’s problem has a consistently negative answer ([16]), Maharam’s problem can be stated as a 1 statement and is therefore, by Shoenfield’s theorem, absolute between transitive models of set theory containing all countable ordinals. Theorem 0.1. Let I be a c.c.c. -ideal on Borel subsets of 2 ! that is analytic on G . The following are equivalent: