Geometric properties of the range of two-dimensional quasi-measures with respect to Radon-Nikodym property

The starting point of this paper are some previous results concerning the existence of a Radon-Nikodym derivative (or equivalently, a density) for a pair of finitely additive measures m and μ, satisfying a suitable absolute continuity condition; in particular in the strongly non atomic case, it was known that if the range of the pair (m,μ) is closed, such a density exist. In the paper it is shown that this condition is just sufficient, but not necessary. Actually the paper precisely describes the planar sets that could in fact be the range of a pair of finitely additive measures (m,μ), with μ<< m, admitting a Radon Nikodym derivative.