On a measure-theoretic area formula

We show how classical differentiation theorems for measures can be turned into an integral representation of a Borel measure with respect to a fixed Carath\'eodory measure. We focus our attention on the cases where this measure is both the Hausdorff measure and the spherical Hausdorff measure, giving the corresponding measure-theoretic area formula. Our point consists in using certain covering derivatives as "generalized densities". Some consequences for the sub-Riemannian Heisenberg group are also pointed out.