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Proceedings of the American Mathematical Society

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On $ \Game\mathbf{\Gamma}$-complete sets


Authors: Gabriel Debs and Jean Saint Raymond
Journal: Proc. Amer. Math. Soc. 148 (2020), 859-873
MSC (2010): Primary 03E15, 28A05; Secondary 54H05
DOI: https://doi.org/10.1090/proc/14731
Published electronically: August 7, 2019
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Abstract: Extending a result of A. Kechris we prove that under suitable assumptions on the class $ \mathbf {\Gamma }$ of Borel sets, in particular when $ \mathbf {\Gamma }$ is a Baire class, if any $ \Game \mathbf {\Gamma }$ set is reducible to some $ \Game \mathbf {\Gamma }$ set $ A$ by a $ \Game \mathbf {\Gamma }$-measurable function, then $ A$ is $ \Game \mathbf {\Gamma }$-complete.


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Additional Information

Gabriel Debs
Affiliation: Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France – and – Université Le Havre Normandie, Institut Universitaire de Technologie, Rue Boris Vian, BP 4006 76610 Le Havre, France
Email: gabriel.debs@imj-prg.fr

Jean Saint Raymond
Affiliation: Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France
Email: jean.saint-raymond@imj-prg.fr

DOI: https://doi.org/10.1090/proc/14731
Keywords: $\LL$-hard, $\LL$-complete, game operator
Received by editor(s): October 8, 2018
Received by editor(s) in revised form: May 29, 2019
Published electronically: August 7, 2019
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2019 American Mathematical Society