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GOLDSTERN’S PRINCIPLE ABOUT UNIONS OF NULL SETS
Part of:
Set theory
Published online by Cambridge University Press: 10 December 2025
Abstract
Goldstern showed in [7] 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 
$\boldsymbol {\Sigma }^1_1$. Our aim is to study to what extent we can drop the 
$\boldsymbol {\Sigma }^1_1$ assumption. We show that Goldstern’s principle for the pointclass 
$\boldsymbol {\Pi }^1_1$ 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.
MSC classification
Primary:
03E15: Descriptive set theory
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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