  In this paper, we study the structural properties of Nevanlinna measures,
i.e. Borel measures that arise in the integral representation of
Herglotz-Nevanlinna functions. In particular, we give a characterization of
these measures in terms of their Fourier transform, characterize measures
supported on hyperplanes including extremal measures, describe the structure of
the singular part of the measures when some variable are set to a fixed value,
and provide estimates for the measure of expanding and shrinking cubes.
Corresponding results are stated also in the setting of the polydisc where
applicable, and some of our proofs are actually perfomed via the polydisc.
