Classifying invariant $\sigma$-ideals with analytic base on good Cantor measure spaces

Abstract

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu (U)<\mu (V)$ there is a clopen set $W\subset V$ with $\mu (W)=\mu (U)$. We study $\sigma$-ideals with Borel base on $X$ which are invariant under the action of the group $H_{\mu }(X)$ of measure-preserving homeomorphisms of $(X,\mu )$, and show that any such $\sigma$-ideal $\mathcal{I}$ is equal to one of seven $\sigma$-ideals: $\{\varnothing \}$, $[X{]}^{\le \omega }$, $\mathcal{E}$, $\mathcal{M}\cap \mathcal{N}$, $\mathcal{M}$, $\mathcal{N}$, or $[X{]}^{\le \mathfrak{c}}$. Here $[X{]}^{\le \kappa }$ is the ideal consisting of subsets of cardiality $\le \kappa$ in $X$, $\mathcal{M}$ is the ideal of meager subsets of $X$, $\mathcal{N}=\{A\subset X:\mu (A)=0\}$ is the ideal of null subsets of $(X,\mu )$, and $\mathcal{E}$ is the $\sigma$-ideal generated by closed null subsets of $(X,\mu )$.

10 pages

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• Everywhere $\mathcal{I}$ sets