Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

Mathias absoluteness and the Ramsey property

In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

On the measurability of functions of two variables

Let f be a real-valued function of two real variables. It is well known(5) that the Lebesgue measurability or even upper semicontinuity of all the sections fx and fy does not imply the measurability of f. Lipiński proved (3) that there exists a non-measurable function f such that each section fx and fy is Baire one, has the Darboux property, and fails to be approximately continuous at no more than one point. Davies (1) noted that the continuum hypothesis implies the existence of a non-measurable function f, such that each fy is measurable and each fx approximately continuous. Ursell (6) proved that if each fy is measurable and each fx continuous (or monotonic), then f is measurable.