Solution of Laczkovich's Lipschitz mapping problem

In 1991, Laczkovich asked whether every measurable set of positive measure in $\mathbb R^d$ can be mapped onto a cube with a Lipschitz map. We show that the answer is positive for every $d$. Our results also give a positive answer for Kolmogorov's problem on contractions onto polygons with small measure loss in the case of Jordan measurable sets.