Extending Baire property by uncountably many sets.

Abstract

We show that for an uncountable κ in a suitable Cohen real model for any family { A_ν} _{ν <kappa} of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A_ν into the algebra of Baire subsets of 2^κ modulo meager sets such that for all Borel B,

B is meager iff h(B)=0.

The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.