Absolute Null Subsets of the Plane with Bad Orthogonal Projections

Abstract

Under Martin’s Axiom, it is proved that there exists an absolute null subset of the Euclidean plane $\mathbb{R}^2$, the orthogonal projections of which on all straight lines in $\mathbb{R}^2$ are absolutely nonmeasurable. A similar but weaker result holds true within the framework of ZFC set theory.