  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.
