  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 $\varepsilon$. 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.
