  Goldstern showed in his 1993 paper that the union of a real-parametrized,
increasing 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 $\mathsf{CH}$ implies the negation of Goldstern's principle for the
pointclass of all subsets. Moreover we show that Goldstern's principle for the
pointclass of all subsets is consistent with $\mathsf{ZFC}$. Also we prove that
Goldstern's principle for the pointclass of all subsets holds both under
$\mathsf{ZF} + \mathsf{AD}$ and in Solovay models.
