When is a Riesz distribution a complex measure?

TL;DR

This paper provides a straightforward proof of the exact conditions under which the Riesz distribution on a Euclidean Jordan algebra is a locally finite complex measure, clarifying its measure-theoretic properties.

Contribution

It offers an elementary proof of the necessary and sufficient conditions for Riesz distributions to be complex measures on Euclidean Jordan algebras.

Findings

Characterization of when R_eta is a complex measure

Elementary proof technique for measure conditions

Clarification of measure-theoretic properties of Riesz distributions

Abstract

Let R_\alpha be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by the complex number \alpha. I give an elementary proof of the necessary and sufficient condition for R_\alpha to be a locally finite complex measure (= complex Radon measure).