We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathe'odory measure. We focus our attention on the case this measure is the spherical Hausdorff measure, giving a metric measure 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.
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