On Borel maps, calibrated ${\sigma }$-ideals, and homogeneity
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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.
References
- Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
- Ryszard Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR 1363947
- Ilijas Farah and Jindřich Zapletal, Four and more, Ann. Pure Appl. Logic 140 (2006), no. 1-3, 3–39. MR 2224046, DOI 10.1016/j.apal.2005.09.002
- Bernard R. Gelbaum, Cantor sets in metric measure spaces, Proc. Amer. Math. Soc. 24 (1970), 341–343. MR 254201, DOI 10.1090/S0002-9939-1970-0254201-7
- W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen (A), Fund. Math., 12 (1928), 78–109.
- Vladimir Kanovei, Marcin Sabok, and Jindřich Zapletal, Canonical Ramsey theory on Polish spaces, Cambridge Tracts in Mathematics, vol. 202, Cambridge University Press, Cambridge, 2013. MR 3135065, DOI 10.1017/CBO9781139208666
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- A. S. Kechris, A. Louveau, and W. H. Woodin, The structure of $\sigma$-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263–288. MR 879573, DOI 10.1090/S0002-9947-1987-0879573-9
- Sabine Koppelberg, Handbook of Boolean algebras. Vol. 1, North-Holland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. MR 991565
- John C. Oxtoby, Homeomorphic measures in metric spaces, Proc. Amer. Math. Soc. 24 (1970), 419–423. MR 260961, DOI 10.1090/S0002-9939-1970-0260961-1
- Roman Pol, Note on Borel mappings and dimension, Topology Appl. 195 (2015), 275–283. MR 3414891, DOI 10.1016/j.topol.2015.09.034
- R. Pol and P. Zakrzewski, On Borel mappings and $\sigma$-ideals generated by closed sets, Adv. Math. 231 (2012), no. 2, 651–663. MR 2955187, DOI 10.1016/j.aim.2012.05.020
- R. Pol and P. Zakrzewski, On Boolean algebras related to $\sigma$-ideals generated by compact sets, Adv. Math. 297 (2016), 196–213. MR 3498798, DOI 10.1016/j.aim.2016.04.010
- Jean Pollard, On extending homeomorphisms on zero-dimensional spaces, Fund. Math. 67 (1970), 39–48. MR 270348, DOI 10.4064/fm-67-1-39-48
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- Marcin Sabok, Forcing, games and families of closed sets, Trans. Amer. Math. Soc. 364 (2012), no. 8, 4011–4039. MR 2912443, DOI 10.1090/S0002-9947-2012-05404-3
- Marcin Sabok and Jindřich Zapletal, Forcing properties of ideals of closed sets, J. Symbolic Logic 76 (2011), no. 3, 1075–1095. MR 2849260, DOI 10.2178/jsl/1309952535
- Sławomir Solecki, Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), no. 3, 1022–1031. MR 1295987, DOI 10.2307/2275926
- Jindřich Zapletal, Forcing idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, Cambridge, 2008. MR 2391923, DOI 10.1017/CBO9780511542732
- Jindřich Zapletal, Descriptive set theory and definable forcing, Mem. Amer. Math. Soc. 167 (2004), no. 793, viii+141. MR 2023448, DOI 10.1090/memo/0793
- Jindřich Zapletal, Dimension theory and forcing, Topology Appl. 167 (2014), 31–35. MR 3193422, DOI 10.1016/j.topol.2014.03.004
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