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: $\{\emptyset\}$, $[X]^{\le\omega}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal N$, 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 $(X,\mu)$.