Some properties of sequential topologies and normal integrands
type: Item
instance of: scholarly article
title: Some properties of sequential topologies and normal integrands
zbMATH Open document ID: 0537.49003
DOI: 10.1007/BF01773499
author: Ada Bottaro Aruffo
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1983-01-01T00:00:00Z
review text: Let \((T,\tau)\), \((T_ i,\tau_ i)\), \((X,\rho)\), \((X_ j,\rho_ j)\) be topological spaces (\(i\in I\), \(j\in J)\), \({\mathcal L}_ i \sigma\)-algebra on \(T_ i\), \({\mathcal L} \sigma\)-algebra on T, \(\mu_ i:{\mathcal L}_ i\to [0,\infty]\), \(\mu: {\mathcal L}\to [0,\infty]\) measures, \({\mathcal H}_{i,j}\subset {\mathcal L}_ i\), \({\mathcal H}\subset {\mathcal L}\), \(\psi_ i:T_ i\to T\), \(\phi_ j:X_ j\to X\), \(E\subset T\times X\), \(E_{i,j}=(\psi_ i\times \phi_ j)^{-1}(E)\), \(f:E\to [- \infty,\infty]\), \(f_{i,j}:E_{i,j}\to [-\infty,\infty]\) such that \[ f_{i,j}=f{\mathbb{O}}((\psi_ i\times \phi_ j)|_{E_{i,j}})\quad(i\in I,\quad j\in J). \] In {\S}2 the author gives conditions to obtain: (a) if \(I(f)=\{t\in T:f(t,.)\) is a lower semicontinuous [resp. sequentially lower semicontinuous] function\}\( \in {\mathcal L}\) and \( \mu(T\backslash I(f))=0\) then \(I(f_{i,j})\in {\mathcal L}_ i\) and \(\mu_ i(T_ i\backslash I(f_{i,j}))=0\) for every \(i\in I\), \(j\in J\); (b) if f is a (\({\mathcal L}\times {\mathcal B}(\rho))/_ E\)- measurable function then \(f_{i,j}\) is a \(({\mathcal L}_ i\times {\mathcal B}(\rho_ j))/_{E_{i,j}}\)-measurable function for every \(i\in I\), \(j\in J\); (c) if for every \(n\in {\mathbb{N}}\) there exists \(K_ n\in {\mathcal K}\) such that \(\mu(T\backslash K_ n)<1/(n+1)\) and \(f/_{E\cap(K_ n\times X)}\) is a lower semicontinuous [resp. sequentially lower semicontinuous] function, then the same condition is true for \(f_{i,j}\) and \({\mathcal H}_{i,j}\) for every \(i\in I\), \(j\in J.\) In {\S}2 the author also gives conditions to obtain the converse of (a), (b), (c). Such results will be utilized by the author in the paper ''Su alcune estensioni del teorema di Scorza-Dragoni'' (to appear), where extensions of Scorza Dragoni's theorem concerning a Lusin type property on normal integrands are proved (\textit{G. Scorza-Dragoni}'s theorem [Rend. Semin. Mat. Univ. Padova 17, 102-106 (1948; Zbl 0032.197)] concerns the case in wh…
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 3853841
zbMATH Keywords: Scorza Dragoni's theorem
zbMATH Keywords: normal integrands
zbMATH Keywords: sequential topologies
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3853841