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) ⊥.