Goldstern's principle about unions of null sets

TL;DR

This paper investigates the extent of Goldstern's principle regarding unions of null sets, extending it to broader pointclasses and analyzing its consistency within various set-theoretic frameworks.

Contribution

It proves Goldstern's principle for the pointclass , shows its consistency with ZFC, and explores its negation under CH and its validity under +AD and Solovay models.

Findings

Goldstern's principle holds for  pointclass.

It is consistent with ZFC.

Negation follows from CH.

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 $\boldsymbol{\Sigma}^1_1$. Our aim is to study to what extent we can drop the $\boldsymbol{\Sigma}^1_1$ assumption. We show 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.