In this paper, we study a class of Borel measures on [Formula: see text] 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 [Formula: see text]-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.

中文翻译：

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与具有非负虚部或实部的全纯函数相关的勒贝格测度和乘积测度的表征

在本文中，我们研究了 [公式：见正文] 上的一类 Borel 测度，它是 Herglotz-Nevanlinna 函数的表示测度类。特别是，我们研究了这个类中的产品度量，其中带有勒贝格度量的产品发挥了特殊作用。因此，我们给出了所有此类测度中[公式：见文本]维勒贝格测度的几种表征，并表征了此类测度中出现的所有乘积测度。此外，还给出了具有消失的混合傅里叶系数的单位多环面上的正Borel测度类的类似结果，并讨论了两类测度与所得结果的关系。