Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates Augusto Ponce Primary 28A78Secondary 35B6035A2035F05Removable singularitydivergenceHausdorff measureHausdorff contentRadon measureFrostman\'s lemmachargesstrong charges We prove that every closed set which is not $\sigma$-finite with respect to the Hausdorff measure $\mathcal{H}^{N-1}$ carries singularities of continuous vector fields in $\mathbb{R}^N$ for the divergence operator. We also show that finite measures which do not charge sets of $sigma$-finite Hausdorff measure $\mathcal{H}^{N-1}$ can be written as an $L^1$ perturbation of the divergence of a continuous vector field. The main tool is a property of approximation of measures in terms of the Hausdorff content. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5079 10.1512/iumj.2013.62.5079 en Indiana Univ. Math. J. 62 (2013) 1055 - 1074 state-of-the-art mathematics http://iumj.org/access/