Three problems of Aronszajn in measure theory
type: Item
instance of: scholarly article
title: Three problems of Aronszajn in measure theory
zbMATH Open document ID: 0576.28003
DOI: 10.1007/BF01086165
publication date: 1984-01-01T00:00:00Z
review text: In the paper the solution of three problems appearing in: \textit{N. Aronszajn}, Stud. Math. 57, 147-190 (1976; Zbl 0342.46034), is given. One of the problems is: to describe all the measures which are absolutely continuous with respect to \({\mathcal U}\{a_ n\}\) for a given sequence \(\{a_ n\}.\) We clarify the necessary notions: \(X\) denotes a separable Banach space, \(B(X)\) the \(\sigma\)-algebra of Borel sets of \(X\). For a given sequence \(\{a_ n\}\) of elements of X the set \({\mathcal U}\{a_ n\}\) denotes the set of such \(B\in B(X)\) for which \(B=\cup_{n}B_ n\), where \(B_ n\in B(X)\) and the usual Lebesgue measure of \((B_ n+x)\cap R^ la_ n\) is equal to zero. Here \(R^ la_ n\) is the line determined by \(a_ n\). The absolute continuity of a measure \(\mu\) with respect to \({\mathcal U}\{a_ n\}\) means that \(\mu\) vanishes on every \(A\in {\mathcal U}\{a_ n\}\).
Mathematics Subject Classification ID: 28A15
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28C99
zbMATH DE Number: 3920844
zbMATH Keywords: differentiability of a measure
zbMATH Keywords: absolute continuity
author: Vladimir I. Bogachev
MaRDI profile type: Publication
cites work: Q4127232
cites work: Q3254057
full work available at URL: https://doi.org/10.1007/bf01086165
OpenAlex ID: W2027805393
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 3920844