  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.
