Constructing diffeomorphisms and homeomorphisms with prescribed derivative

We prove that for any measurable mapping T into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals T outside a set of measure less than ε. We use this fact to prove that for any measurable mapping T into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals T almost everywhere.