More Tales of Two (s)-ities
Kenneth E. Schilling · Real Analysis Exchange · 2005
In any complete separable metric space, the Boolean algebra $(s)/(s_0)$ of Marczewski sets modulo the Marczewski null sets is complete. Using this fact, we show that a simple construction solves two known problems in real analysis: the existence of an $(s_0)$-set which is not Lebesgue measurable 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.