On a bad descriptive structure of Minkowski’s sum of certain small sets in a topological vector space

Abstract

For some natural classes of topological vector spaces, we show the absolute nonmeasurability of Minkowski’s sum of certain two universal measure zero sets. This result can be considered as a strong form of the classical theorem of Sierpinski [8] stating the existence of two Lebesgue measure zero subsets of the Euclidean space, whose Minkowski’s sum is not Lebesgue measurable.