Lipschitz functions with prescribed blowups at many points

In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure μ . We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at μ -almost every point x in any direction which is not contained in the decomposability bundle V ( μ , x ) , recently introduced by Alberti and the first author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point x if it is null at the origin and it is the sum of a linear function on V ( μ , x ) and a Lipschitz function on V ( μ , x ) ⊥ .