On Vitali sets and their unions

Abstract

It is well known that any Vitali set on the real line $\mathbb{R}$ does not possessthe Baire property. In this article we prove the following:Let $S$ be a Vitali set, $S_r$ be the image of $S$ under thetranslation of $\mathbb{R}$ by a rational number $r$ and $\mathcal{F}\mathcal{=}\{{\mathcal{S}}_{\mathcal{r}}\mathcal{:}\mathcal{r}\mathcal{\text{ is rational}}\}$. Then for each non-empty propersubfamily ${\mathcal{F}}^{\prime }$ of $\mathcal{F}$ the union $\bigcup {\mathcal{F}}^{\prime }$ does notpossess the Baire property.