On Vitali sets and their unions

Abstract

It is well known that any Vitali set on the real line $\Bbb{R}$ 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 $\Bbb {R}$ by a rational number $r$ and $\Cal F= \{S_r: r \text{ is rational}\}$. Then for each non-empty propersubfamily $\Cal F'$ of $\Cal F$ the union $\bigcup \Cal F'$ does notpossess the Baire property.