Can we assign the Borel hulls in a monotone way?

• Autorzy: Márton Elekes , András Máthé
• Języki publikacji: EN

Abstrakty

EN

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].

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 03E17: Cardinal characteristics of the continuum
• 28A51: Lifting theory
• 28E15: Other connections with logic and set theory
• 03E15: Descriptive set theory
• 03E35: Consistency and independence results
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 205
• Numer: 2
• Strony: 105-115

Daty

wydano

2009

Twórcy

autor

Márton Elekes

• Rényi Alfréd Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary

autor

András Máthé

• Eötvös Loránd University, Department of Analysis, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary

Identyfikatory

DOI

10.4064/fm205-2-2