Can we assign the Borel hulls in a monotone way?

Márton Elekes¹ ; András Máthé²

¹ Rényi Alfréd Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127 H-1364 Budapest, Hungary
² Eötvös Loránd University Department of Analysis Pázmány Péter sétány 1//c H-1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 205 (2009) no. 2, pp. 105-115

Résumé

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 $[0,1]$.

Détail

DOI : 10.4064/fm205-2-2

Keywords: hull subseteq set containing lambda * lambda * investigate versions following problem does there exist monotone respect inclusion map assigns borel delta hull every negligible measurable subset three versions turn out independent zfc while fourth only prove nonexistence monotone delta hull operation measurable sets consistent remains whether existence here consistent answer question nbsp gyenes nbsp lgyi whether monotone hulls defined every chain measurable sets moreover comment problem hulls subsets