  We consider the approximation of Poisson type problems where the source is
given by a singular measure and the domain is a convex polygonal or polyhedral
domain. First, we prove the well-posedness of the Poisson problem when the
source belongs to the dual of a weighted Sobolev space where the weight belongs
to the Muckenhoupt class. Second, we prove the stability in weighted norms for
standard finite element approximations under the quasi-uniformity assumption on
the family of meshes.
