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 $\mathcal {H}_\mu (X)$ 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 cardinality $\le \kappa$ 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 $(X,\mu )$.