Goldstern's Principle with respect to Hausdorff Measures

TL;DR

This paper investigates Goldstern's principle in relation to Hausdorff measures, revealing its failure in the constructible universe $L$ for certain sets, but its validity under the assumption of a measurable cardinal.

Contribution

It demonstrates the failure of the Hausdorff measure version of Goldstern's principle in $L$ and establishes its consistency with the existence of a measurable cardinal.

Findings

Hausdorff measure version fails in $L$ for $oldsymbol{oldsymbol{ ext{Pi}}}^1_1$ sets

Lebesgue measure version of Goldstern's principle holds in $L$

Goldstern's principle holds if a measurable cardinal exists

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 $\boldsymbol{\Pi}^1_1$ 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.