Rational normalization of concentration measures

Abstract

We study normalization features of good concentration measures. We extend the classical normalization to the case in which one requires a linear dependence between rational fractions of occupation and values of the concentration measure between 0 and 1. This is called rational normalization. This principle is studied for general good concentration measures. In this context, a stability property of good concentration measures is proved. It is also shown that every good concentration measure can be rationally normalized. For the concentration measures of Theil and Atkinson, explicit calculations of the rational normalization are given. Furthermore, we show that, amongst the generalized Pratt measures, the original Pratt measure is the only one to be rationally normalized.