Sul rango di una massa vettoriale

This paper focuses on the structure of the range of a strongly non atomic (i.e. strongly bounded) vector-valued finitely additive measures, in both the finite dimensional and the infinite dimensional case. In the finite-dimensional case we prove that the Lyapounoff Theorem has to be weakened in the finitely additive case; more precisely the range is always convex, but, as an example shows, it is not closed (and therefore not compact) in general. On the contrary, in the infinite dimensional case, where even in the countably additive case one just describes the weak or strong closure of the range, the use of the Stone extension and a density argument lead to conclusion completely similar to those of Bartle-Dunford-Schwartz, Kluvanek, Uhl.