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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Classifying invariant $\sigma$-ideals with analytic base on good Cantor measure spaces
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by Taras Banakh, Robert Rałowski and Szymon Żeberski PDF
Proc. Amer. Math. Soc. 144 (2016), 837-851 Request permission

Abstract:

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu (U)<\mu (V)$ there is a clopen set $W\subset V$ with $\mu (W)=\mu (U)$. We study $\sigma$-ideals with Borel base on $X$ which are invariant under the action of the group $\mathcal {H}_\mu (X)$ of measure-preserving homeomorphisms of $(X,\mu )$, and show that any such $\sigma$-ideal $\mathcal {I}$ is equal to one of seven $\sigma$-ideals: $\{\emptyset \}$, $[X]^{\le \omega }$, $\mathcal E$, $\mathcal {M}\cap \mathcal N$, $\mathcal {M}$, $\mathcal N$, or $[X]^{\le \mathfrak c}$. Here $[X]^{\le \kappa }$ is the ideal consisting of subsets of cardinality $\le \kappa$ in $X$, $\mathcal {M}$ is the ideal of meager subsets of $X$, $\mathcal N=\{A\subset X:\mu (A)=0\}$ is the ideal of null subsets of $(X,\mu )$, and $\mathcal E$ is the $\sigma$-ideal generated by closed null subsets of $(X,\mu )$.
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Additional Information
  • Taras Banakh
  • Affiliation: Department of Mathematics, Ivan Franko National University of Lviv, Ukraine — and — Institute of Mathematics, Jan Kochanowski University, Kielce, Poland
  • MR Author ID: 249694
  • Email: t.o.banakh@gmail.com
  • Robert Rałowski
  • Affiliation: Faculty of Fundamental Problems of Technology, Institute of Mathematics and Computer Science, Wrocław University of Technology, Wrocław, Poland
  • MR Author ID: 362782
  • Email: robert.ralowski@pwr.wroc.pl
  • Szymon Żeberski
  • Affiliation: Faculty of Fundamental Problems of Technology, Wrocław University of Technology, Wrocław, Poland
  • Email: szymon.zeberski@pwr.wroc.pl
  • Received by editor(s): September 13, 2014
  • Received by editor(s) in revised form: January 3, 2015
  • Published electronically: October 7, 2015
  • Additional Notes: This work was partially financed by NCN means granted by decision DEC-2011/01/B/ST1/ 01439.
  • Communicated by: Mirna Dzamonja
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 837-851
  • MSC (2010): Primary 03E15, 28A05, 28D05, 54H05
  • DOI: https://doi.org/10.1090/proc/12709
  • MathSciNet review: 3430858