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

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