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We prove that every closed set which is not sigma-finite with respect to theHausdorff measure H^{N-1} carries singularities of continuous vector fields inthe Euclidean space R^N for the divergence operator. We also show that finitemeasures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} canbe written as an L^1 perturbation of the divergence of a continuous vectorfield. The main tool is a property of approximation of measures in terms of theHausdorff content.

Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates