TY - JOUR
AU - Nyagahakwa, Venuste
AU - Haguma, Gratien
PY - 2021/11/30
Y2 - 2022/01/22
TI - Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 127
IS - 3
SE - Articles
DO - 10.7146/math.scand.a-128969
UR - https://www.mscand.dk/article/view/128969
SP -
AB - <p>In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.</p>
ER -