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

## TL;DR

This paper investigates the homogeneity properties of certain $\sigma$-ideals related to Borel measures on compact spaces, revealing non-homogeneity in some cases and potential homogeneity in others, with implications for measure theory and topology.

## Contribution

It demonstrates that the $\sigma$-ideal $I(dim)$ on the Hilbert cube is strongly non-homogeneous and explores conditions under which quotient Boolean algebras become homogeneous.

## Key findings

- $I(dim)$ is not homogeneous on the Hilbert cube
- Quotients by $J_0(\mu)$ or $J_f(\mu)$ can be homogeneous
- Discusses calibrated $\sigma$-ideals in a general setting

## 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.