On Borel hull operations

TL;DR

This paper investigates the existence of Borel hull operations under certain set-theoretic assumptions, showing their non-existence for null and meager sets and linking monotonicity properties to the property of Baire.

Contribution

It demonstrates the non-existence of translation invariant Borel hull operations under Martin's Axiom and connects monotone Borel hull operations to the property of Baire.

Findings

No translation invariant Borel hull operation exists under Martin's Axiom.

Monotone Borel hull operation on meager sets implies one on sets with the property of Baire.

Set-theoretic assumptions influence the structure of Borel hull operations.

Abstract

We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.