Characterizations of the Lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part

TL;DR

This paper characterizes Lebesgue and product measures within the class of representing measures for holomorphic functions with non-negative imaginary or real parts, providing new insights into their structure and relations.

Contribution

It offers new characterizations of Lebesgue measure and product measures among representing measures of Herglotz-Nevanlinna functions, linking measures on Euclidean space and the unit polytorus.

Findings

Characterization of Lebesgue measure among representing measures.

Description of product measures within the class.

Relations between measures on Euclidean space and the polytorus.

Abstract

In this paper, we study a class of Borel measures on $\mathbb{R}^n$ that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the $n$-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.