Orders of absolute measurability
type: Item
instance of: scholarly article
title: Orders of absolute measurability
zbMATH Open document ID: 0965.28001
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-07-17T00:00:00Z
review text: The author considers absolutely measurable subsets of the torus \([0,1)^k\) and various questions concerning equidecomposability of such sets. \(A\) is absolutely measurable if the value of \(\mu(A)\) is the same for every Banach measure \(\mu\) (i.e., a finitely additive and translation-invariant probability measure defined on all subsets of \([0,1)^k\)). He defines four set-functions (called orders): the Jordan order \(o(A)\), the absolute order \(\delta(A)\), the Tarski order \(\tau(A)\) and the scissor order \(\sigma(A)\), and shows that for every set \(A\subset[0,1)^k\): \(o(A)\ll \delta(A)\ll\tau(a)\ll\sigma(A)\). (Here \(\alpha\ll\beta\) means that there is a positive constant \(c\) only depending on \(k\) such that \(\alpha\leq c\cdot\beta\).) Finally, the author establishes relationships between orders of a set \(A\) and equivalence of \(A\) to a cube. For example: If \(A\) is equivalent to a cube then \(\delta(A)>0\) and \(\tau(A)=\sigma(A)=\infty\). If \(\lambda_k(A)>0\) and \(o(A)>0\) then \(A\) is equivalent to a cube. If \(\delta(A)=\infty\) then \(A\) is equivalent to a cube.
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 28E15
zbMATH DE Number: 1545041
zbMATH Keywords: absolutely measurable set
zbMATH Keywords: Banach measure
zbMATH Keywords: Jordan measurability
zbMATH Keywords: absolute inner measure
zbMATH Keywords: absolute outer measure
zbMATH Keywords: equidecomposability
zbMATH Keywords: Jordan order
zbMATH Keywords: absolute order
zbMATH Keywords: Tarski order
zbMATH Keywords: scissor order
author: M. Laczkovich
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7031
OpenAlex ID: W2066288835
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DOI: 10.1006/JMAA.2000.7031
description: scientific article; zbMATH DE number 1545041