On the notions of upper and lower density

Let be the power set of N. We say that a function is an upper density if, for all X, Y † N and h, k N+, the following hold: (f1); (f2) if X † Y; (f3); (f4), where k · X: = {kx: x X}; and (f5). 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 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.