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Can we assign the Borel hulls in a monotone way?

Tom 205 / 2009

Márton Elekes, András Máthé Fundamenta Mathematicae 205 (2009), 105-115 MSC: Primary 28A51; Secondary 03E15, 03E17, 03E35, 28A05, 28E15, 54H05. DOI: 10.4064/fm205-2-2

Streszczenie

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 $[0,1]$.

Autorzy

  • Márton ElekesRényi Alfréd Institute of Mathematics
    Hungarian Academy of Sciences
    P.O. Box 127
    H-1364 Budapest, Hungary
    e-mail
  • András MáthéEötvös Loránd University
    Department of Analysis
    Pázmány Péter sétány 1//c
    H-1117 Budapest, Hungary
    e-mail

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