GOLDSTERN’S PRINCIPLE ABOUT UNIONS OF NULL SETS

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.