Goldstern's principle about unions of null sets

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 ${\mathbf{\Sigma }}_1^1$. Our aim is to study to what extent we can drop the ${\mathbf{\Sigma }}_1^1$ assumption. We show Goldstern’s principle for the pointclass ${\mathbf{\Pi }}_1^1$ holds. We show that Goldstern’s principle for the pointclass of all subsets is consistent with $\mathsf{Z}\mathsf{F}\mathsf{C}$ and show its negation follows from $\mathsf{C}\mathsf{H}$. Also we prove that Goldstern’s principle for the pointclass of all subsets holds both under $\mathsf{Z}\mathsf{F}+\mathsf{A}\mathsf{D}$ and in Solovay models.