LIPSCHITZ IMAGE OF A MEASURE-NULL SET CAN.pdf

ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria Lipschitz Image of a Measure{null SetCan Have a Null ComplementJ. LindenstraussE. MatouskovaD. Preiss Vienna, Preprint ESI 736 (1999) August 5, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via anonymous ftp or gopher from FTP.ESI.AC.ATor via WWW, URL: http://www.esi.ac.at LIPSCHITZ IMAGE OF A MEASURE-NULL SET CANHAVE A NULL COMPLEMENTJ. LINDENSTRAUSS, E. MATOUSKOVA, AND D. PREISSAbstract. We give two examples that in innite dimensional Ba-nach spaces the measure-null sets are not preserved by Lipschitzhomeomorphisms. There exists a closed set D  `2 which containsa translate of any compact set in the unit ball of `2 and a Lips-chitz isomorphism F of `2 onto `2 so that F (D) is contained in ahyperplane. Let X be a Banach space with an unconditional basis.There exists a Borel set A  X and a Lipschitz isomorphism F ofX onto itself so that the sets X nA and F (A) are both Haar null.1. IntroductionIn innite dimensional Banach spaces there is no natural analogue ofLebesgue null sets. Various measure-theoretic notions of null sets wereintroduced there in order to generalize the theorem of Rademacherto innite dimensional Banach spaces: if f is a Lipschitz function ona separable Banach space then it is Ga?teaux dierentiable \almosteverywhere (see [BL] for a survey of the topic). More generally, iff is a Lipschitz mapping from a separable Banach space to a Banachspace with the Radon-Nikodym property (re exive Banach spaces arean example), then it is Ga?teaux dierentiable \almost everywhere.Let X and Y be separable, and, say, re exive Banach spaces, let F :X ! Y be a Lipschitz isomorphism; that is, F is surjective and bothF and F1 are Lipschitz. It is an open question if there is an x 2 X sothat the Ga?teaux derivative of F at x is surjective. If the answer waspositive, Lipschitz isomorphic separable re exive Banach spaces