Extended Gini Index

We propose an extended version of Gini index defined on the set of infinite utility streams, $X=Y^\mathbb{N}$ where $Y\subset \mathbb{R}$. For $Y$ containing at most finitely many elements, the index satisfies the generalized Pigou-Dalton transfer principles in addition to the anonymity axiom. However, in the general case of $Y$ containing infinitely many elements, real valued representation satisfying the generalized Pigou-Dalton transfer principle and anonymity is not possible.