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Lipschitz image of a measure-null set can have a null complement |
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| Authors: | J Lindenstrauss E Matoušková D Preiss |
| Institution: | (1) Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel;(2) Mathematical Institute, Czech Academy of Sciences, Žitná 25, CZ-11567 Prague, Czech Republic;(3) Department of Mathematics, University College London, Gower Street, WC1E 6BT London, UK |
| 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 ? ?2 which contains a translate of any compact set in the unit ball of ?2 and a Lipschitz isomorphismF of ?2 onto ?2 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. |
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