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$F_\sigma$-additive families and the invariance of Borel classes

Author: Jirí Spurny
Journal: Proc. Amer. Math. Soc. 133 (2005), 905-915
MSC (2000): Primary 54H05, 54E40; Secondary 28A05
Published electronically: September 20, 2004
MathSciNet review: 2113943
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Abstract: We prove that any $F_\sigma$-additive family $\mathcal{A}$ of sets in an absolutely Souslin metric space has a $\sigma$-discrete refinement provided every partial selector set for $\mathcal{A}$ is $\sigma$-discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps $F_\sigma$-sets to $F_\sigma$-sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves $F_\sigma$-sets, is shown as an application of the previous result.

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Additional Information

Jirí Spurny
Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Keywords: $F_\sigma$--additive family, $\sigma$--discrete refinement, first class selector, Borel classes
Received by editor(s): April 10, 2003
Received by editor(s) in revised form: October 30, 2003
Published electronically: September 20, 2004
Additional Notes: This research was supported in part by the grant GAČR 201/03/0935, GAČR 201/03/D120 and in part by the Research Project MSM 1132 00007 from the Czech Ministry of Education
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

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