Kechris, A. S. and Solecki, S. (1995) Approximation of analytic by Borel sets and definable countable chain conditions. Israel Journal of Mathematics, 89 (1-3). pp. 343-356. ISSN 0021-2172. doi:10.1007/BF02808208. https://resolver.caltech.edu/CaltechAUTHORS:20130521-103730289
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Abstract
Let I be a σ-ideal on a Polish space such that each set from I is contained in a Borel set from I. We say that I fails to fulfil the Σ_1^1 countable chain condition if there is a Σ_1^1 equivalence relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of Borel sets from I is definable in the codes of Borel sets, then each Σ_1^1 set is equal to a Borel set modulo a set from I iff I fulfils the Σ_1^1 countable chain condition. Further we characterize the σ-ideals I generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for Σ_1^1 sets mentioned above. It turns out that they are exactly of the form MGR(F)={A : ∀F ∈ F A ∩F is meager in F} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal on R, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.
Item Type: | Article | |||||||||
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Additional Information: | © 1995 Springer-Verlag. Received July 20, 1993 and in revised form March 6, 1994. Research partially supported by NSF grant DMS-9317509. | |||||||||
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Issue or Number: | 1-3 | |||||||||
DOI: | 10.1007/BF02808208 | |||||||||
Record Number: | CaltechAUTHORS:20130521-103730289 | |||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20130521-103730289 | |||||||||
Official Citation: | Approximation of analytic by Borel sets and definable countable chain conditions A. S. Kechris, S. Solecki pp.343-356 | |||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
ID Code: | 38600 | |||||||||
Collection: | CaltechAUTHORS | |||||||||
Deposited By: | Ruth Sustaita | |||||||||
Deposited On: | 21 May 2013 17:54 | |||||||||
Last Modified: | 09 Nov 2021 23:38 |
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