## 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