On measurability of algebraic sums of small sets

It is shown that, under Martin’s Axiom, the algebraic sum of two universal measure zero subsets of the real line can be an absolutely nonmeasurable set. Some related questions concerning measurability of algebraic sums of small sets are also discussed.

On Absolutely Nonmeasurable Sets and Functions

We consider some properties of sets and functions which are measurable (or nonmeasurable) with respect to certain classes of measures. In this context, the notion of an absolutely nonmeasurable set (function) is examined. Sierpiński-Zygmund type functions are constructed having additional properties closely connected with the above-mentioned notion. Also, some small subsets of uncountable commutative groups are discussed whose algebraic sum turns out to be an absolutely nonmeasurable set.

On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑2 which are 𝑇2-negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇2 denotes the group of all translations of 𝐑2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇2, where 𝑔 is an arbitrary rotation of 𝐑2 distinct from the identity transformation and all central symmetries of 𝐑2.