The Dynkin system generated by the large balls of \(\mathbb R^n\). (English) Zbl 0970.28002

Let \(X\) be a nonempty set. We say that a system \(\mathcal D\) of subsets of \(X\) is a Dynkin system if \(\mathcal D\) contains \(X\) and is closed with respect to complements and countable disjoint unions. The author proves that the smallest Dynkin system in \(\mathbb R^n\) containing all open balls with radii \(\geq 1\) does not contain all Borel subsets of \(\mathbb R^n\). Moreover, the characterization of the elements of this Dynkin system is given. The presented method was used by the author and D. Preiss to prove that the smallest Dynkin system containing all open balls of an infinite-dimensional separable Hilbert space does not contain all Borel sets [T. Keleti and D. Preiss, Math. Proc. Camb. Philos. Soc. 128, No. 3, 539-547 (2000; Zbl 0952.28001)]
Reviewer’s remark: Theorem 9 is true only for uncountable sets.


28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E15 Descriptive set theory


Zbl 0952.28001