## The Banach-Sachs property and Haar null sets.(English)Zbl 0937.46011

The author modifies the definition of Haar null set in an equivalent way and uses it to prove the following fact.
Let $$X$$ be a Banach space such that $$X^*$$ has the Banach-Sachs property (i.e. every bounded sequence $$(x_n^*)\subset X^*$$ has a subsequence $$(x_{n_k}^*)$$ with the norm convergent Cesaro means $$\frac 1k \sum _i^kx^*_{n_i}$$) and let $$K\subset X$$ be a convex closed set with empty interior. Then there is a probability measure $$\mu$$ on $$X$$ such that $$\mu (K+x)=0$$ for all $$x\in X$$.
This may not be true if the Haar nullness is replaced by Aronszajn nullness as is noticed at the end of the paper. Namely, any closed convex set $$K$$ in a separable Banach space $$X$$ satisfying $$\overline {\text{span}} K=X$$ cannot be Aronszajn null set.

### MSC:

 46B20 Geometry and structure of normed linear spaces 28A75 Length, area, volume, other geometric measure theory

### Keywords:

Banach space; convexity; Haar null set
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