Lipschitz image of a measure-null set can have a null complement

Abstract

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ⊂ l2 which contains a translate of any compact set in the unit ball of l2 and a Lipschitz isomorphismF of l2 onto l2 so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA⊂X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.