A. Kharazishvili

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 ${\bf R}^2$ which are $T_2$-negligible and, simultaneously, $G$-absolutely nonmeasurable. Here $T_2$ denotes the group of all translations of ${\bf R}^2$ and $G$ denotes the group generated by $\{g\} \cup T_2$, where $g$ is an arbitrary rotation of ${\bf R}^2$ distinct from the identity transformation and all central symmetries of ${\bf R}^2$.

Sierpinski partition, quasi-invariant measure, uniform set, negligible set, absolutely nonmeasurable set.

MSC 2000: 28A05, 28D05