The strength of measurability hypotheses
type: Item
instance of: scholarly article
title: The strength of measurability hypotheses
zbMATH Open document ID: 0602.03012
DOI: 10.1007/BF02759764
author: Jean Raisonnier
author: Jacques Stern
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Using ideas from the proof of Shelah's theorem, it is shown that if every set in the (boldface) \(\Sigma^ 1_ 2\) hierarchy is measurable, then these sets have the Baire property, as well. With similar reasoning, it is shown that if the union of \(\kappa\) many measure-zero sets is always a measure-zero set, then the union of \(\kappa\) meager sets is meager, a result independently proved by Bartoszynski. If for every family of measure-zero sets of size less than continuum, there is a measure-zero set not covered by any member in the family, then this holds for meager, as well. Finally, it is shown that the perfect set theorem is implied by the following statement: whenever \(A_ x\) is nonempty for \(x\in B\), B of positive measure, then there exists a Borel-function f such that \(f(x)\in A_ x\) for a.e. \(x\in B\).
reviewed by: Péter Komjáth
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 3970741
zbMATH Keywords: Baire property
zbMATH Keywords: measure-zero sets
zbMATH Keywords: meager sets
zbMATH Keywords: perfect set theorem
zbMATH Keywords: Borel-function
MaRDI profile type: Publication
cites work: Additivity of Measure Implies Additivity of Category
cites work: Additivity of Measure Implies Dominating Reals
cites work: Q5620774
cites work: A mathematical proof of S. Shelah's theorem on the measure problem and related results
cites work: Q3342556
cites work: Q5515834
cites work: Can you take Solovay's inaccessible away?
cites work: A model of set-theory in which every set of reals is Lebesgue measurable
cites work: Some measure theoretic results in effective descriptive set theory
cites work: Regularity properties of definable sets of reals
full work available at URL: https://doi.org/10.1007/bf02759764
OpenAlex ID: W2044117691
description: scientific article; zbMATH DE number 3970741