A generalized Vitali set from nonextensive statistics

TL;DR

This paper generalizes the Vitali set using nonextensive statistics and a $q$-addition, exploring its measure-theoretic properties across different parameter values, revealing non-measurability for most cases.

Contribution

It introduces a novel generalization of the Vitali set based on nonextensive algebra and analyzes its measure-theoretic properties across the parameter space.

Findings

Vitali set is non-measurable for rational $q$ in (1/2, 1]

Measurability cannot be guaranteed at $q o 1/2$

If measurable at $q o 1/2$, then measure must be positive

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 $\frac{1}{2}<q\leq1$, but in the limit $q\rightarrow\frac{1}{2}$ the non-measurability cannot be guaranteed. Furthermore, assuming measurability when $q\rightarrow\frac{1}{2}$, then this must be positive.