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

For every finite measure μ on R^n we define a decomposability bundle V(μ, ·) related to the decompositions of μ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on R^n is differentiable at μ-a.e. x with respect to the subspace V(μ, x), and prove that this differentiability result is optimal, in the sense that, following [4], we can construct Lipschitz functions which are not differentiable at μ-a.e. x in any direction which is not in V(μ,x). 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 basic formulas involving normal currents and maps of class C^1 to Lipschitz maps.