Density sets of sets of positive integers

Extending previous results, we give a new description of the density set, that is the set of all pairs of densities – upper and lower – of all subsets of a given set of positive integers. The extension consists in using the concept of weighted density with the weight function satisfying two standard conditions. In order to prove that the density set is convex, we establish and use the joint Darboux property of the weighted density. Finally we prove that the density set is closed through an explicit characterization of its upper boundary.