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Goldstern's Principle with respect to Hausdorff Measures

Tatsuya Goto

TL;DR

The paper extends Goldstern's principle to the context of Hausdorff measures, showing that the Hausdorff-measure version of GP for $\boldsymbol{\Pi}^1_1$ sets fails in $L$ but can hold under the existence of a measurable cardinal. It develops a forcing- and capacity-based framework to analyze $\mathsf{GP}$ across doubling gauge functions, with a main dichotomy between non-$\sigma$-finite and $\sigma$-finite cases, and uses Solovay's measure uniformization to derive $\mathsf{GP}({\text{all}})$. Further results relate $\mathsf{GP}$ to cardinal invariants (e.g., $\operatorname{cov}(\mathcal{M})$, $\mathfrak{b}$) and show implications such as $\mathsf{GP}({\text{all}})$ forcing $\mathcal{SN}=\mathcal{NA}$, as well as demonstrating failures for certain closed-$\mathcal{E}$ variants. The discussion outlines open questions and the interaction between large cardinals, regularity properties, and geometric-measure-theoretic GP principles, highlighting the landscape of $GP$ under different models and pointclasses.

Abstract

This paper is a continuation of the paper [Got25] and studies Goldstern's principle, a principle about unions of continuum many null sets, further. The main result is that the Hausdorff measure version of Goldstern's principle for $\boldsymbolΠ^1_1$ sets fails in $L$, despite the fact that the Lebesgue measure version is true. Moreover, we show that this version holds provided that the measurable cardinal exists. Other various results regarding Goldstern's principle are established.

Goldstern's Principle with respect to Hausdorff Measures

TL;DR

The paper extends Goldstern's principle to the context of Hausdorff measures, showing that the Hausdorff-measure version of GP for sets fails in but can hold under the existence of a measurable cardinal. It develops a forcing- and capacity-based framework to analyze across doubling gauge functions, with a main dichotomy between non--finite and -finite cases, and uses Solovay's measure uniformization to derive . Further results relate to cardinal invariants (e.g., , ) and show implications such as forcing , as well as demonstrating failures for certain closed- variants. The discussion outlines open questions and the interaction between large cardinals, regularity properties, and geometric-measure-theoretic GP principles, highlighting the landscape of under different models and pointclasses.

Abstract

This paper is a continuation of the paper [Got25] and studies Goldstern's principle, a principle about unions of continuum many null sets, further. The main result is that the Hausdorff measure version of Goldstern's principle for sets fails in , despite the fact that the Lebesgue measure version is true. Moreover, we show that this version holds provided that the measurable cardinal exists. Other various results regarding Goldstern's principle are established.
Paper Structure (11 sections, 21 theorems, 27 equations, 1 table)

This paper contains 11 sections, 21 theorems, 27 equations, 1 table.

Table of Contents

  1. Introduction
  2. The countable ideal version of $\mathsf{GP}$
  3. The Hausdorff measure version of $\mathsf{GP}$
  4. Miscellaneous results on $\mathsf{GP}$ for Lebesgue measure
  5. $\mathsf{GP}$ for pointclasses between $\boldsymbol{\Pi}^1_1$ and $\boldsymbol{\Delta}^1_2$
  6. Solovay's measure uniformization and $\mathsf{GP}(\mathsf{all})$
  7. New necessary condition for $\mathsf{GP}(\mathsf{all})$
  8. New sufficient condition for $\mathsf{GP}(\boldsymbol{\Pi}^1_2)$
  9. $\mathsf{GP}(\mathsf{all})$ implies that strong measure zero ideal equals null additive ideal
  10. $\mathsf{GP}(\mathsf{closed}, \mathcal{E})$ fails
  11. Discussions

Key Result

Lemma 1.5

Let $X$ be a doubling metric space and $f$ be a gauge function. Then there is a $c > 0$ such that for every $\delta > 0$ and $A \subseteq X$ we have

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • Proposition 1.10
  • Proposition 2.1
  • proof
  • ...and 37 more