Classifying invariant $σ$-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 $σ$-additive measure $μ$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $μ(U)<μ(V)$ there is a clopen set $W\subset V$ with $μ(W)=μ(U)$. We study $σ$-ideals with Borel base on $X$ which are invariant under the action of the group $H_μ(X)$ of measure-preserving homeomorphisms of $(X,μ)$, and show that any such $σ$-ideal $\mathcal I$ is equal to one of seven $σ$-ideals: $\{\emptyset\}$, $[X]^{\leω}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal N$, or $[X]^{\le \mathfrak c}$. Here $[X]^{\leκ}$ is the ideal consisting of subsets of cardiality $\leκ$ in $X$, $\mathcal M$ is the ideal of meager subsets of $X$, $\mathcal N=\{A\subset X:μ(A)=0\}$ is the ideal of null subsets of $(X,μ)$, and $\mathcal E$ is the $σ$-ideal generated by closed null subsets of $(X,μ)$.