Can we assign the Borel hulls in a monotone way?

TL;DR

This paper explores the possibility of defining monotone hull operations for measurable sets in [0,1], revealing independence results from ZFC and addressing open questions about hulls of chains and all subsets.

Contribution

It investigates the existence and independence of monotone hull operations for measurable sets, providing new insights into set-theoretic and measure-theoretic properties.

Findings

Three versions are independent of ZFC.

Nonexistence of a monotone G_delta hull is consistent.

Open question on the existence of such hulls for all measurable sets.

Abstract

A \emph{hull} of $A \subset [0,1]$ 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 (wrt. 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 (the usual Zermelo-Fraenkel axioms with the Axiom of Choice), 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 a question of Z. Gyenes and D. P\'alv\"olgyi which asks if monotone hulls can be defined for every chain (wrt. inclusion) of measurable sets. We also comment on the problem of hulls of all subsets of $[0,1]$.