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Transactions of the American Mathematical Society

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On Borel maps, calibrated $ {\ensuremath{\sigma}}$-ideals, and homogeneity


Authors: R. Pol and P. Zakrzewski
Journal: Trans. Amer. Math. Soc. 370 (2018), 8959-8978
MSC (2010): Primary 03E15, 54H05; Secondary 28A78, 54F45
DOI: https://doi.org/10.1090/tran/7462
Published electronically: August 31, 2018
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Abstract: Let $ \mu $ be a Borel measure on a compactum $ X$. The main objects in this paper are $ {\ensuremath {\sigma }}$-ideals $ I(\dim )$, $ J_0(\mu )$, $ J_f(\mu )$ of Borel sets in $ X$ that can be covered by countably many compacta which are finite-dimensional, or of $ \mu $-measure null, or of finite $ \mu $-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the $ {\ensuremath {\sigma }}$-ideal $ I(\dim )$ is not homogeneous in a strong way. We shall also show that in some natural instances of measures $ \mu $ with nonhomogeneous $ {\ensuremath {\sigma }}$-ideals $ J_0(\mu )$ or $ J_f(\mu )$, the completions of the quotient Boolean algebras $ {\rm Borel}(X)/J_0(\mu )$ or $ {\rm Borel}(X)/J_f(\mu )$ may be homogeneous.

We discuss the topic in a more general setting, involving calibrated $ {\ensuremath {\sigma }}$-ideals.


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Additional Information

R. Pol
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Email: pol@mimuw.edu.pl

P. Zakrzewski
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Email: piotrzak@mimuw.edu.pl

DOI: https://doi.org/10.1090/tran/7462
Keywords: Borel mapping, ${\ensuremath{\sigma}}$-ideal, meager set, infinite dimension, Borel measure
Received by editor(s): June 27, 2017
Received by editor(s) in revised form: November 10, 2017
Published electronically: August 31, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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