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The balls do not generate all Borel sets using complements and countable disjoint unions. (English) Zbl 0952.28001

A nonempty family \(\mathcal D\) of subsets of a set \(X\) is said to be a Dynkin system if \(\mathcal D\) is closed under complements and countable unions. For a metric space \(X\) let \(\mathcal D(X)\) denote the Dynkin system generated by the family of all open balls in \(X\). Clearly, \(\mathcal D(X)\) is a subclass of the class \(\mathcal B(X)\) of all Borel subsets of \(X\). The motivation for the question whether \(\mathcal D(X)=\mathcal B(X)\) comes from the fact that two Borel probability measures which agree on open balls agree on \(\mathcal D(X)\). R. O. Davies [Mathematika 18, 157-160 (1971; Zbl 0229.28005)] constructed a compact metric space with two different Borel probability measures on \(P\) agreeing on all balls. The question whether this is true was open although D. Preiss and J. Tišer [Mathematika 38, No. 2, 391-397 (1991; Zbl 0755.28006)] have proved that in any separable Banach space two different measures cannot agree on all open balls. The authors of the paper under review prove that the equality \(\mathcal D(X)=\mathcal B(X)\) cannot be proved from the Preiss-Tišer result and in fact that \(\mathcal D(X)\neq\mathcal B(X)\) for every separable infinite dimensional Hilbert space \(X\).

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54E99 Topological spaces with richer structures
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