The strength of measurability hypotheses
Using ideas from the proof of Shelah's theorem, it is shown that if every set in the (boldface) \(\Sigma^ 1_ 2\) hierarchy is measurable, then these sets have the Baire property, as well. With similar reasoning, it is shown that if the union of \(\kappa\) many measure-zero sets is always a measure-zero set, then the union of \(\kappa\) meager sets is meager, a result independently proved by Bartoszynski. If for every family of measure-zero sets of size less than continuum, there is a measure-zero set not covered by any member in the family, then this holds for meager, as well. Finally, it is shown that the perfect set theorem is implied by the following statement: whenever \(A_ x\) is nonempty for \(x\in B\), B of positive measure, then there exists a Borel-function f such that \(f(x)\in A_ x\) for a.e. \(x\in B\).
scientific article; zbMATH DE number 3970741