A note on the Sierpiński partition
type: Item
instance of: scholarly article
zbMATH Open document ID: 0872.28002
DOI: 10.1515/JAA.1996.41
author: Alexander Kharazishvili
published in: Journal of Applied Analysis
publication date: 1997-10-05T00:00:00Z
full work available at URL: https://eudml.org/doc/47704
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 981720
zbMATH Keywords: non-measurable set
zbMATH Keywords: set without the property of Baire
zbMATH Keywords: Sierpiński partition
zbMATH Keywords: Dieudonné measure
MaRDI profile type: Publication
OpenAlex ID: W2087040623
title: A note on the Sierpiński partition
review text: For the Sierpiński partition \(A=\{(\xi,\zeta): \xi\leq\zeta<\omega_1\}\), \(B=\{(\xi,\zeta):\omega_1>\xi>\zeta\}\) of \(\omega_1\times\omega_1\) and the Dieudonné measure \(\lambda\) on \(\omega_1\) it is shown that \(A\), \(B\) are non-measurable with respect to \(\lambda\otimes\lambda\) in (ZF)\&(DC), whereas it is impossible to prove in (ZF)\&(DC) the existence of a non-measurable subset of \(\omega_1\) with respect to \(\lambda\). Furthermore, in (ZF)\&(DC) both sets \(A\) and \(B\) have the property of Baire in the product space \((\omega_1,\tau)\times (\omega_1,\tau)\) for a suitable topology \(\tau\) of \(\omega_1\), whereas the existence of a subset of \((\omega_1,\tau)\) without the Baire property cannot be established in (ZF)\&(DC). Finally, it is proved that in (ZF)\&(DC) one cannot conclude from the existence of some subset of a topological space without the property of Baire that there exists a subset of the corresponding topological product space without the property of Baire. However, one can show in (ZF)\&(DC) that the existence of a non-measurable subset with respect to some \(\sigma\)-finite measure \(\mu\) implies the existence of some non-measurable subset of the corresponding measure-theoretical product with respect to \(\mu\otimes\mu\).
description: scientific article; zbMATH DE number 981720