Lipschitz image of a measure-null set can have a null complement (Q1580506)

In finite dimensions, a Lipschitz map takes null sets to null sets. Here the authors provide the following infinite-dimensional counterexamples for the class of Haar null sets: (1) There exists a closed set \(D\subset \ell_{2}\) which contains a translate of any compact set in the unit ball of \(\ell_{2}\) and a Lipschitz isomorphism \(F\) of \(\ell_{2}\) onto \(\ell_{2}\) so that \(F(D)\) is contained in a hyperplane. (2) In every infinite-dimensional Banach space \(X\) with an unconditional basis there exist a Borel subset \(A\) and a Lipschitz isomorphism of \(X\) onto itself so that the sets \(X\setminus A\) and \(F(A)\) are both Haar null. The authors also prove that -- unlike the finite-dimensional case -- in an infinite-dimensional space any two nets are Lipschitz equivalent.