On the notions of upper and lower density

Abstract

Let $\mathcal{P}(\mathbf{N})$ be the power set of $\mathbf{N}$. We say that a function ${\mu }^{\ast }:\mathcal{P}(\mathbf{N})\to \mathbf{R}$ is an upper density if, for all $X,Y\subseteq \mathbf{N}$ and $h,k\in {\mathbf{N}}^{+}$, the following hold: (F1) ${\mu }^{\ast }(\mathbf{N})=1$; (F2) ${\mu }^{\ast }(X)\le {\mu }^{\ast }(Y)$ if $X\subseteq Y$; (F3) ${\mu }^{\ast }(X\cup Y)\le {\mu }^{\ast }(X)+{\mu }^{\ast }(Y)$; (F4) ${\mu }^{\ast }(k\cdot X)=\frac{1}{k}{\mu }^{\ast }(X)$, where $k\cdot X:=\{kx:x\in X\}$; (F5) ${\mu }^{\ast }(X+h)={\mu }^{\ast }(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $\alpha$-densities (with $\alpha$ a real parameter $\ge -1$), 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 $X\subseteq \mathbf{N}$. 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.

26 pp, no figs. Added a 'Note added in proof' at the end of Sect. 7 to answer Question 6. Final version to appear in Proc. Edinb. Math. Soc. (the paper is a prequel of arXiv:1510.07473)

References in corpus (3)

• Lévy group and density measures
• Extreme points of the set of density measures
• Upper and lower densities have the strong Darboux property

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