Inserted: 9 jan 2014
Last Updated: 13 jan 2021
Journal: Proc. Roy. Soc. Edinburgh Sect. A
We show how classical differentiation theorems for measures can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory 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.