A general setting for the pointwise investigation of determinacy

Abstract

It is well-known that if we assume a large class of sets of reals to be determined then we may conclude that all sets in this class have certain regularity properties: we say that determinacy implies regularity properties classwise. In [Lö05] the pointwise relation between determinacy and certain regularity properties (namely the Marczewski-Burstin algebra of arboreal forcing notions and a corresponding weak version) was examined. An open question was how this result extends to topological forcing notions whose natural measurability algebra is the class of sets having the Baire property. We study the relationship between the two cases, and using a definition which adequately generalizes both the Marczewski-Burstin algebra of measurability and the Baire property, prove results similar to [Lö05]. We also show how this can be further generalized for the purpose of comparing algebras of measurability of various forcing notions. © 2009 Springer Berlin Heidelberg.