**A. Kharazishvili**

## On Negligible and Absolutely Nonmeasurable
Subsets of the Euclidean Plane

**Abstract:**

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 ${\bf R}^2$.

**Keywords:**

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

**MSC 2000:** 28A05, 28D05