某类Hausdorff测度的可加性可与Lebesgue测度的可加性不同
The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure
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中文总结 AI 辅助
本文通过Tukey归约将Davies-Rogers Hausdorff测度零理想与局部化系统关联，证明其可加性和共尾性可与Lebesgue零理想不同。
中文摘要 AI 辅助
设$\mathcal{N}^h_\Omega$为Davies和Rogers构造的Hausdorff测度的零理想。我们给出了$(\mathcal{N}^h_\Omega,\mathcal{N}^h_\Omega,\subseteq)$到具有有限坐标集的局部化系统的Tukey归约。由此可得$\mathfrak{v}^\forall_{D,g}\le\operatorname{add}(\mathcal{N}^h_\Omega)$且$\operatorname{cof}(\mathcal{N}^h_\Omega)\le\mathfrak{c}^\forall_{D,g}$。因此，我们证明了$\mathfrak{d}<\operatorname{add}(\mathcal{N}^h_\Omega)$以及分别地$\operatorname{cof}(\mathcal{N}^h_\Omega)<\mathfrak{b}$的一致性。从而$\mathcal{N}^h_\Omega$的可加性和共尾性可与Lebesgue零理想的不同。
英文摘要
Let $\mathcal{N}^h_Ω$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $(\mathcal{N}^h_Ω,\mathcal{N}^h_Ω,\subseteq)$ to a localization system with finite coordinate sets. It follows that $\mathfrak{v}^\forall_{D,g}\le\operatorname{add}(\mathcal{N}^h_Ω)$ and $\operatorname{cof}(\mathcal{N}^h_Ω)\le\mathfrak{c}^\forall_{D,g}$ for some $D, g \in ω^ω$. Consequently, we prove the consistency of $\mathfrak{d}<\operatorname{add}(\mathcal{N}^h_Ω)$ and, separately, $\operatorname{cof}(\mathcal{N}^h_Ω)<\mathfrak{b}$. Hence the additivity and cofinality of $\mathcal{N}^h_Ω$ 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 $\operatorname{cov}(\mathcal{N}^h_Ω)\le\operatorname{non}(\mathcal{N}^h_Ω)$.
发表机构
- TU Wien（维也纳工业大学）
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