Let $\mathcal {P}(\mathbf{N})$ be the power set of N. We say that a function $\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$ is an upper density if, for all X, Y ⊆ N and h, k ∈ N+, the following hold: (f1) $\mu ^\ast (\mathbf{N}) = 1$; (f2) $\mu ^\ast (X) \le \mu ^\ast (Y)$ if X ⊆ Y; (f3) $\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$; (f4) $\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$, where k · X : = {kx: x ∈ X}; and (f5) $\mu ^\ast (X + h) = \mu ^\ast (X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)–(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that $\mu ^\ast (X)\le 1$ for every X ⊆ N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

中文翻译：

---

关于上密度和下密度的概念

让$\mathcal {P}(\mathbf{N})$成为的幂集ñ. 我们说一个函数$\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$是一个上密度，如果，对于所有X,是⊆ñ和H,ķ∈ñ+，以下成立：（f1)$\mu ^\ast (\mathbf{N}) = 1$; (f2)$\mu ^\ast (X) \le \mu ^\ast (Y)$如果X⊆是; (f3)$\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$; (f4)$\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$， 在哪里ķ·X: = {kx：X∈X}; 和 （f5)$\mu ^\ast (X + h) = \mu ^\ast (X)$. 我们证明了上渐近、上对数、上 Banach、上 Buck、上 Pólya 和上解析密度，以及所有上α-密度（与αa real parameter ≥ -1)，是我们定义意义上的上密度。此外，我们建立了公理的相互独立性（f1)–(f5)，并且我们在假设 (f2) 替换为较弱的条件$\mu ^\ast (X)\le 1$对于每个X⊆ñ. 总的来说，这使我们能够扩展和推广迄今为止独立推导出的一些经典上密度的结果，从而在理论中引入了一定程度的统一。