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

Roman Pol; Piotr Zakrzewski

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arXiv · 1706.04773

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

Roman Pol·Piotr Zakrzewski

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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 non-homogeneous $\sigma$-ideals $J_0(\mu)$ or $J_f(\mu)$, the completions of the quotient Boolean algebras $Borel(X)/J_0(\mu)$ or $Borel(X)/J_f(\mu)$ may be homogeneous. We discuss the topic in a more general setting, involving calibrated $\sigma$-ideals.

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BibTeX RIS

Roman Pol, Piotr Zakrzewski. 2017-06-15. On Borel maps, calibrated $\sigma$-ideals and homogeneity. https://arxiv.org/abs/1706.04773

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