  Abstract upper densities are monotone and subadditive functions from the
power set of positive integers to the unit real interval that generalize the
upper densities used in number theory, including the upper asymptotic density,
the upper Banach density, and the upper logarithmic density. We answer a
question posed by G. Grekos in 2013, and prove the existence of translation
invariant abstract upper densities onto the unit interval, whose null sets are
precisely the family of finite sets, or the family of sequences whose series of
reciprocals converge. We also show that no such density can be atomless. (More
generally, these results also hold for a large class of summable ideals.)
