Some subsets of the Euclidean plane

Abstract

Here we would like to consider those subsets of the Euclidean plane R₂ which possess rather strange geometrical and measure-theoretical properties. In particular, we will be dealing with thick (massive) subsets of R₂ having small linear sections in all possible directions, and we will investigate these sets from the point of view of the measurability with respect to translation-invariant extensions of the classical Lebesgue measure λ2 on R₂.