More Tales of Two (s)-ities.

Abstract: In any complete separable metric space, the Boolean algebra (s)/(s0) of Marczewski sets modulo the Marczewski null sets is complete. Using this fact, we show that a simple construction solves two known prob-lems in real analysis: the existence of an (s)0-set which is not Lebesgue mea-surable and does not have the Baire property, and a function which is not (s)-measurable, but whose graph is an (s)0-set. Using a similar idea, we also present a short proof that the Boolean algebra of universally measurable sets modulo the sets universally of measure zero is not complete. Let X and Y be complete separable metric spaces. In [S], we constructed by a fairly simple method an (s)-set in X × Y which does not belong to the σ-algebra generated by products A×B of (s)-sets A ⊂ X and B ⊂ Y. The key idea was to apply the fact that (s)/(s0) is a complete Boolean algebra, and to find a collection of (s)-sets whose least upper bound in that algebra has the desired properties. Here we show that this same construction can also be used to give short solutions to other known problems in the theor