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

  • Tatsuya Goto
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Abstract

Let 𝒩hΩ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of (𝒩hΩ,𝒩hΩ,⊆) to a localization system with finite coordinate sets. It follows that v∀D,g≤add(𝒩hΩ) and cof(𝒩hΩ)≤c∀D,g for some D, g ∈ ωω. Consequently, we prove the consistency of d<add(𝒩hΩ) and, separately, cof(𝒩hΩ)<b. Hence the additivity and cofinality of 𝒩hΩ 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 cov(𝒩hΩ)≤non(𝒩hΩ).

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Cite

BibTeXRIS
BibTeX
@misc{goto2026additivity,
  title = {{The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure}},
  author = {Tatsuya Goto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08549},
  eprint = {2610.08549},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08549},
}
RIS
TY  - UNPB
TI  - The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure
AU  - Tatsuya Goto
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.08549
UR  - https://arxiv.org/abs/2610.08549
AB  - Let 𝒩^h_Ω be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of (𝒩^h_Ω,𝒩^h_Ω,⊆) to a localization system with finite coordinate sets. It follows that v^∀_D,g≤add(𝒩^h_Ω) and cof(𝒩^h_Ω)≤c^∀_D,g for some D, g ∈ ω^ω. Consequently, we prove the consistency of d<add(𝒩^h_Ω) and, separately, cof(𝒩^h_Ω)<b. Hence the additivity and cofinality of 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 cov(𝒩^h_Ω)≤non(𝒩^h_Ω).
ER  -