The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure

Abstract

Let ${\mathcal{N}}_{\Omega }^h$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $({\mathcal{N}}_{\Omega }^h,{\mathcal{N}}_{\Omega }^h,\subseteq )$ to a localization system with finite coordinate sets. It follows that ${\mathfrak{v}}_{D,g}^{\forall }\le \mathrm{add}({\mathcal{N}}_{\Omega }^h)$ and $\mathrm{cof}({\mathcal{N}}_{\Omega }^h)\le {\mathfrak{c}}_{D,g}^{\forall }$ for some $D,g\in {\omega }^{\omega }$. Consequently, we prove the consistency of $\mathfrak{d}<\mathrm{add}({\mathcal{N}}_{\Omega }^h)$ and, separately, $\mathrm{cof}({\mathcal{N}}_{\Omega }^h)<\mathfrak{b}$. Hence the additivity and cofinality of ${\mathcal{N}}_{\Omega }^h$ can differ from those of the Lebesgue null ideal. We also show that the finite graphs in the Davies--Rogers construction can be chosen so that $\mathrm{cov}({\mathcal{N}}_{\Omega }^h)\le \mathrm{non}({\mathcal{N}}_{\Omega }^h)$.

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