On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

Abstract: Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $\mu$. In particular we show that the differentiability properties of Lipschitz functions at $\mu$-almost every point are related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.