This paper deals with a generalization of a result of Geometric Measure Theory: if a rectifiable set is closed, then its Minkowski content is equal to its Hausdorff measure. This result is generalized to an anisotropic and continuous perturbation of the Hausdorff measure. So, an adapted Minkowski content is introduced and the proof that it is equal to the perturbed Hausdorff measure under the same hypothesis of closure and rectifiability is given. This result is applied to the jump set of a minimizer of a free boundary problem for the case where the Hausdorff measure of the boundary is perturbed by an anisotropic and continuous function and a rigorous setting is given for the approximation of an anisotropic version for the Mumford-Shah functional.