On a property of Herglotz functions

Abstract

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be the singular part of the measure $\mu_r$ which corresponds to $g_r(z)$ via the Herglotz representation theorem. Then the measure $\int_0^1 \mu_r^{(s)}\,dr$ is absolutely continuous, its density is integer-valued a.e., and moreover the density takes values $0$ or $1$ a.e.