A generalized Vitali set from nonextensive statistics

Abstract

We address a generalization of the Vitali set through a deformed translational property that stems from a generalized algebra derived from the nonextensive statistics. The generalization is based on the so-called $q$-addition $x{\oplus }_{q}y=x+y+(1-q)xy$ for rational values of $q$, where the ordinary formalism is recovered when the control parameter $q\to 1$. The generalized Vitali set is non-measurable for all rational parameter $\frac{1}{2}<q\le 1$, but in the limit $q\to \frac{1}{2}$ the non-measurability cannot be guaranteed. Furthermore, assuming measurability when $q\to \frac{1}{2}$, then this must be positive.