A refined Lusin type theorem for gradients

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field f coincides with the gradient of a $C^1$ function g, outside a set E of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure μ, and we obtain that the estimate on the $L^p$ norm of Dg does not depend on $\mu \left(E\right)$, if the value of f is μ-a.e. orthogonal to the decomposability bundle of μ. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $R^n$ and we state a suitable generalization for k-forms, which would imply the validity of the conjecture in full generality.