Can we assign the Borel hulls in a monotone way?

Márton Elekes; András Máthé

Fundamenta Mathematicae (2009)

  • Volume: 205, Issue: 2, page 105-115
  • ISSN: 0016-2736

Abstract

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A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(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 δ 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 δ 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].

How to cite

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Márton Elekes, and András Máthé. "Can we assign the Borel hulls in a monotone way?." Fundamenta Mathematicae 205.2 (2009): 105-115. <http://eudml.org/doc/286301>.

@article{MártonElekes2009,
abstract = {A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_\{δ\}$ 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 $G_\{δ\}$ 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].},
author = {Márton Elekes, András Máthé},
journal = {Fundamenta Mathematicae},
keywords = {hull; envelope; Borel; monotone; Lebesgue measure; Cohen real; continuum hypothesis},
language = {eng},
number = {2},
pages = {105-115},
title = {Can we assign the Borel hulls in a monotone way?},
url = {http://eudml.org/doc/286301},
volume = {205},
year = {2009},
}

TY - JOUR
AU - Márton Elekes
AU - András Máthé
TI - Can we assign the Borel hulls in a monotone way?
JO - Fundamenta Mathematicae
PY - 2009
VL - 205
IS - 2
SP - 105
EP - 115
AB - A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{δ}$ 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 $G_{δ}$ 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].
LA - eng
KW - hull; envelope; Borel; monotone; Lebesgue measure; Cohen real; continuum hypothesis
UR - http://eudml.org/doc/286301
ER -

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