In the following paper, one studies, given a bounded, connected open set Ω ⊆ R n , κ > 0, a positive Radon measure µ 0 in Ω and a (signed) Radon measure µ on Ω satisfying µ(Ω) = 0 and |µ| κµ 0 , the possibility of solving the equation div u = µ by a vector field u satisfying |u| κw on Ω (where w is an integrable weight only related to the geometry of Ω and to µ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with measure data, and also construct a weight w that relies in a softer way on the geometry of Ω, improving its behavior (and hence the a priori behavior of the solution we construct) substantially in some instances. The method used in this paper follows a constructive approach of Bogovskii type.