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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On Borel maps, calibrated ${\sigma }$-ideals, and homogeneity
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by R. Pol and P. Zakrzewski PDF
Trans. Amer. Math. Soc. 370 (2018), 8959-8978 Request permission

Abstract:

Let $\mu$ be a Borel measure on a compactum $X$. The main objects in this paper are ${\sigma }$-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 ${\sigma }$-ideal $I(\dim )$ is not homogeneous in a strong way. We shall also show that in some natural instances of measures $\mu$ with nonhomogeneous ${\sigma }$-ideals $J_0(\mu )$ or $J_f(\mu )$, the completions of the quotient Boolean algebras $\textrm {Borel}(X)/J_0(\mu )$ or $\textrm {Borel}(X)/J_f(\mu )$ may be homogeneous.

We discuss the topic in a more general setting, involving calibrated ${\sigma }$-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
  • MR Author ID: 239503
  • Email: piotrzak@mimuw.edu.pl
  • Received by editor(s): June 27, 2017
  • Received by editor(s) in revised form: November 10, 2017
  • Published electronically: August 31, 2018
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 3864401