On Borel maps, calibrated $σ$-ideals and homogeneity

Abstract

Let $μ$ be a Borel measure on a compactum $X$. The main objects in this paper are $σ$-ideals $I(dim)$, $J_0(μ)$, $J_f(μ)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of $μ$-measure null, or of finite $μ$-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the $σ$-ideal $I(dim)$ is not homogeneous in a strong way. We shall also show that in some natural instances of measures $μ$ with non-homogeneous $σ$-ideals $J_0(μ)$ or $J_f(μ)$, the completions of the quotient Boolean algebras $Borel(X)/J_0(μ)$ or $Borel(X)/J_f(μ)$ may be homogeneous. We discuss the topic in a more general setting, involving calibrated $σ$-ideals.