Classifying invariant ▫$\sigma$▫-ideals with analytic base on good Cantor measure spaces
Banakh, Taras, 1968- ; Rałowski, Robert ; Żeberski, Szymon
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 ▫$(X,\mu)$▫.
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