Some properties of sequential topologies and normal integrands
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 which the function is a Carathéodory function). In the first part of the paper some properties of the topology \(s\tau\) are studied, where \(s\tau\) is the strongest topology in which a sequence converges to x if and only if it converges to x in \(\tau\) ; the behaviour of \(s\tau\) is studied with respect to some operations on the topologies (restriction to a subspace, final topology, product). Such results can also be used to simplify some hypotheses of the theorems of {\S}2.
scientific article; zbMATH DE number 3853841