Can we assign the Borel hulls in a monotone way?
Volume 205 / 2009
Abstract
A hull of $A\subseteq[0,1]$ is a set $H$ containing $A$ such that $\lambda^*(H)=\lambda^*(A)$. We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_\delta$ hull to every negligible/measurable subset of $[0,1]$?
Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_\delta$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer the question of Z. Gyenes and D. Pálvölgyi whether monotone hulls can be defined for every chain of measurable sets. Moreover, we comment on the problem of hulls of all subsets of $[0,1]$.
