On a measure which measures at least one selector for every uncountable subgroup

Abstract

We show that there exists in ZFC an invariant extension of Lebesgue measure on \(\mathbb{R}\) such that for every uncountable subgroup \(H\) of \(\mathbb{R}\) there exists at least one selector of \(H\) measurable with respect to this measure. This answers a question of Sławomir Solecki in \cite{S}.