The Dynkin system generated by balls in $\mathbb {R}^d$ contains all Borel sets
HTML articles powered by AMS MathViewer
- by Miroslav Zelený
- Proc. Amer. Math. Soc. 128 (2000), 433-437
- DOI: https://doi.org/10.1090/S0002-9939-99-05507-0
- Published electronically: September 23, 1999
- PDF | Request permission
Abstract:
We show that for every $d \in \mathbb {N}$ each Borel subset of the space $\mathbb {R}^{d}$ with the Euclidean metric can be generated from closed balls by complements and countable disjoint unions.References
- Heinz Bauer, Probability theory and elements of measure theory, Second edition of the translation by R. B. Burckel from the third German edition, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. MR 636091
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Vladimír Olejček, The $\sigma$-class generated by balls contains all Borel sets, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3665–3675. MR 1327035, DOI 10.1090/S0002-9939-1995-1327035-7
- D. Preiss, T. Keleti, The balls do not generate all Borel sets using complements and countable disjoint unions (to appear).
- D. Preiss and J. Tišer, Measures in Banach spaces are determined by their values on balls, Mathematika 38 (1991), no. 2, 391–397 (1992). MR 1147839, DOI 10.1112/S0025579300006744
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Bibliographic Information
- Miroslav Zelený
- Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 186 00, Czech Republic
- Email: zeleny@karlin.mff.cuni.cz
- Received by editor(s): February 11, 1998
- Published electronically: September 23, 1999
- Additional Notes: This research was supported by Research Grant GAUK 190/1996 and GAČR 201/97/1161.
- Communicated by: Frederick W. Gehring
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 433-437
- MSC (1991): Primary 28A05, 04A15
- DOI: https://doi.org/10.1090/S0002-9939-99-05507-0
- MathSciNet review: 1695330