Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-dimensional Lebesgue Measure

We consider some subsets of the Euclidean plane R2, having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on R2. Many examples of paradoxical sets in a finite-dimensional Euclidean space, having small sections by hyperplanes in this space, are known in the literature. One of the earliest examples is due to Sierpiński who constructed a function f : R → R such that its graph is λ2-thick in R2. Here λ2 denotes the standard two-dimensional Lebesgue measure on the plane R2, and we say that a subset X of R2 is λ2-thick in R2 if the inner λ2measure of the set R2 \X is equal to zero (cf. the corresponding definition in [1]). In particular, the λ2-thickness of the graph of f implies that it is nonmeasurable with respect to λ2 and, hence, f is not measurable in the Lebesgue sense. At the same time, any straight line in R2 parallel to the axis of ordinates meets the graph of f in exactly one point. Further, Mazurkiewicz constructed a subset Y of R2 having the property that, for each straight line l in R2, the set l ∩ Y consists of exactly two points. Note that Y can also be chosen to be λ2-thick and, consequently, nonmeasurable with respect to λ2. Later on, various examples of sets with small sections, however large in some sense, were presented by other authors. In this paper, we deal with similar sets in connection with the following natural question: how small are such sets from the point of view of invariant extensions of the Lebesgue measure λ2? Namely, we are going to demonstrate in our further considerations that there are invariant extensions of λ2, concentrated on sets with small linear sections (in all directions). Actually, 1991 Mathematics Subject Classification. 28A05, 28D05.