On the Sierpiński-Erdős and the Oxtoby-Ulam theorems for some new sigma-ideals of sets

Abstract:

Let $\Phi (\Psi )$ denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. There is a homeomorphism of the unit square onto itself mapping a given set in $\Phi (\Psi )$) onto a set of Lebesgue measure zero. Theorem 2. There is a set belonging to both $\Phi$ and $\Psi$ that cannot be mapped onto a set of first category by a homeomorphism of the unit square onto itself. Let C denote the Cantor set, regarded as the product of a sequence of 2-element groups, and let $\Lambda$ denote one of the $\sigma$-ideals of subsets of C studied by Schmidt and Mycielski. Theorem 3. Assuming the continuum hypothesis, the Sierpiński-Erdös theorem holds for $\Lambda$ and the class of subsets of C of Haar measure zero (or of first category). Theorem 4. The Oxtoby-Ulam theorem holds for the image of $\Lambda$ under the Cantor mapping of C onto the unit interval.