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Goldstern's principle about unions of null sets

Tatsuya Goto

TL;DR

The paper investigates how Goldstern’s principle extends beyond the original Σ^1_1 setting by defining GP(Γ) and analyzing its validity across projective levels, determinacy, and Solovay models. It proves GP(Π^1_1) and establishes both consistency and non-consistency results for GP(all): GP(all) is consistent with ZFC and with ZF+AD, and ¬GP(all) is consistent under CH via null towers and forcing arguments. The work further shows GP(all) in Solovay models and delineates determinacy and large-cardinal consequences, including separations among projective GP notions and necessary conditions for GP(Δ^1_2). It also extends the framework to other ideals, providing counterexamples for GP with E and discussing a wide array of open problems. Overall, the paper maps the landscape of when monotone unions of null sets remain null under weakened definability assumptions and clarifies the roles of determinacy, large cardinals, and forcing in these questions.

Abstract

Goldstern showed in his 1993 paper that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbolΣ^1_1$. Our aim is to study to what extent we can drop the $\boldsymbolΣ^1_1$ assumption. We show Goldstern's principle for the pointclass $\boldsymbolΠ^1_1$ holds. We show that Goldstern's principle for the pointclass of all subsets is consistent with $\mathsf{ZFC}$ and show its negation follows from $\mathsf{CH}$. Also we prove that Goldstern's principle for the pointclass of all subsets holds both under $\mathsf{ZF} + \mathsf{AD}$ and in Solovay models.

Goldstern's principle about unions of null sets

TL;DR

The paper investigates how Goldstern’s principle extends beyond the original Σ^1_1 setting by defining GP(Γ) and analyzing its validity across projective levels, determinacy, and Solovay models. It proves GP(Π^1_1) and establishes both consistency and non-consistency results for GP(all): GP(all) is consistent with ZFC and with ZF+AD, and ¬GP(all) is consistent under CH via null towers and forcing arguments. The work further shows GP(all) in Solovay models and delineates determinacy and large-cardinal consequences, including separations among projective GP notions and necessary conditions for GP(Δ^1_2). It also extends the framework to other ideals, providing counterexamples for GP with E and discussing a wide array of open problems. Overall, the paper maps the landscape of when monotone unions of null sets remain null under weakened definability assumptions and clarifies the roles of determinacy, large cardinals, and forcing in these questions.

Abstract

Goldstern showed in his 1993 paper that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly . Our aim is to study to what extent we can drop the assumption. We show Goldstern's principle for the pointclass holds. We show that Goldstern's principle for the pointclass of all subsets is consistent with and show its negation follows from . Also we prove that Goldstern's principle for the pointclass of all subsets holds both under and in Solovay models.
Paper Structure (11 sections, 23 theorems, 51 equations)

This paper contains 11 sections, 23 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Review of Goldstern's proof
  3. $\mathsf{GP}(\boldsymbol{\Pi}^1_1)$
  4. Consistency of $\neg \mathsf{GP}(\mathsf{all})$
  5. Consistency of $\mathsf{GP}(\mathsf{all})$
  6. A necessary condition for $\mathsf{GP}(\boldsymbol{\Delta}^1_2)$
  7. Consequences of determinacy
  8. Consequences of large cardinals
  9. $\mathsf{GP}(\mathsf{all})$ in Solovay models
  10. Other ideals than the Lebesgue measure zero ideal
  11. Open problems

Key Result

Theorem 2.1

Let $(Y, \mu)$ be a Polish probability space. Let $A \subseteq \omega^\omega \times Y$ be a $\boldsymbol{\Sigma}^1_1$ set. Assume that for each $x \in \omega^\omega$, has measure $0$. Also, assume that $(\forall x, x' \in \omega^\omega)(x \le x' \Rightarrow A_{x} \subseteq A_{x'})$. Then $\bigcup_{x \in \omega^\omega} A_x$ also has measure $0$.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: Goldstern
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 3.2
  • ...and 41 more