Convexity and Haar Null Sets

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 μ on X so that μ(C + x) = 0 for all x ∈ 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. A Borel subset A of a separable Banach space X is called a Haar null set if there exists a probability measure μ on the σ-algebra of Borel subsets of X so that μ(A+x) = 0 for all x ∈ X (see [C] also for the following properties of Haar null sets). The family of all such sets is closed under translation and under countable unions; nonempty open sets do not belong to it. The Haar null sets agree with Lebesgue null Borel sets in finite dimensional spaces. This definition of null sets is rather weak, as every compact set in an infinite dimensional space is a Haar null set. In fact, in infinite dimensional superreflexive and nonreflexive Banach spaces even all weakly compact convex sets with empty interior are Haar null. For superreflexive spaces this follows from our result. If K is a weakly compact and convex subset of a nonreflexive Banach space, then ⋃ t>0 t(K −K) 6= X and there exists x ∈ X so that the intersection of the line segment [0, x] and any translate of K contains at most one point. Consequently, if μ is Lebesgue measure on [0, x], then μ(K+x) = 0 for each x ∈ X . In [MS] it is shown that a separable Banach space is nonreflexive if and only if there exists a closed convex subset Q of X with empty interior, which contains a translate of any compact subset of X . Such a set Q is not Haar null because given any probability measure μ on X there exists a compact set K with μ(K) > 0 and, consequently, also a translate of Q of positive measure. It follows that every separable nonreflexive Banach space contains a closed convex set with empty interior which is not Haar null. In this note we show that this is unlike the situation in superreflexive spaces, where for every closed convex set C with empty interior there exists a Radon probability measure μ on X so that μ(C + x) = 0 for all x ∈ X . We do not know if such a measure exists when X is only reflexive or, equivalently, if the positive cone of every reflexive Banach space with a basis is Haar null. Received by the editors February 22, 1995 and, in revised form, January 8, 1996. 1991 Mathematics Subject Classification. Primary 46B10; Secondary 46B20.