@MISC{Farah04betweenmaharam’s, author = {Ilijas Farah}, title = {Between Maharam’s and von Neumann’s problems}, year = {2004} }

Share

OpenURL

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). 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 Σ12 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. σ-ideal on Borel subsets of 2ω that is analytic on Gδ. The following are equivalent: • PI is a weakly distributive notion of forcing • there is a continuous submeasure on 2ω such that I is the σ-ideal of its null sets. A suitable large cardinal assumption implies that the assumption ‘I is analytic on Gδ ’ can be relaxed to ‘I is definable.’