  In this paper, we study the properties of a certain class of Borel measures
on $\mathbb{R}^n$ that arise in the integral representation of
Herglotz-Nevanlinna functions. In particular, we find that restrictions to
certain hyperplanes are of a surprisingly simple form and show that the
supports of such measures can not lie within particular geometric regions, e.g.
strips with positive slope. Corresponding results are derived for measures on
the unit poly-torus with vanishing mixed Fourier coefficients. These measures
are closely related to functions mapping the unit polydisk analytically into
the right half-plane.
