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Convexity and Haar null sets

Author: Eva Matousková
Journal: Proc. Amer. Math. Soc. 125 (1997), 1793-1799
MSC (1991): Primary 46B10; Secondary 46B20
MathSciNet review: 1372040
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Abstract: It is shown that for every closed, convex and nowhere dense subset $C$ of a superreflexive Banach space $X$ there exists a Radon probability measure $\mu $ on $X$ so that $\mu (C+x)=0$ for all $x\in X$. In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.

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Eva Matousková
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83 , 18600 Prague, Czech Republic; Institut für Mathematik, Johannes Kepler Universität, Altenbergerstraße, A-4040 Linz, Austria

Keywords: Superreflexive Banach spaces, convexity, Haar null sets
Received by editor(s): February 22, 1995
Received by editor(s) in revised form: January 8, 1996
Additional Notes: The author was partially supported by the grant GAČR 201/94/0069 and by a grant of the Austrian Ministry of Education.
Communicated by: Dale Alspach
Article copyright: © Copyright 1997 American Mathematical Society