(U\)-topology and \(m\)-topology on the ring of Measurable Functions, generalized and revisited

• Published in: arXiv.org
• Main Authors: Nandi, Pratip, Atasi Deb Ray, Acharyya, Sudip Kumar
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.06.2023
• Subjects: Rings (mathematics); Topology
• ISSN: 2331-8422

Let \(\mathcal{M}(X,\mathcal{A})\) be the ring of all real valued measurable functions defined over the measurable space \((X,\mathcal{A})\). Given an ideal \(I\) in \(\mathcal{M}(X,\mathcal{A})\) and a measure \(\mu:\mathcal{A}\to[0,\infty]\), we introduce the \(U_\mu^I\)-topology and the \(m_\mu^I\)-topology on \(\mathcal{M}(X,\mathcal{A})\) as generalized versions of the topology of uniform convergence or the \(U\)-topology and the \(m\)-topology on \(\mathcal{M}(X,\mathcal{A})\) respectively. With \(I=\mathcal{M}(X,\mathcal{A})\), these two topologies reduce to the \(U_\mu\)-topology and the \(m_\mu\)-topology on \(\mathcal{M}(X,\mathcal{A})\) respectively, already considered before. If \(I\) is a countably generated ideal in \(\mathcal{M}(X,\mathcal{A})\), then the \(U_\mu^I\)-topology and the \(m_\mu^I\)-topology coincide if and only if \(X\setminus \bigcap Z[I]\) is a \(\mu\)-bounded subset of \(X\). The components of \(0\) in \(\mathcal{M}(X,\mathcal{A})\) in the \(U_\mu^I\)-topology and the \(m_\mu^I\)-topology are realized as \(I\cap L^\infty(X,\mathcal{A},\mu)\) and \(I\cap L_\psi(X,\mathcal{A},\mu)\) respectively. Here \(L^\infty(X,\mathcal{A},\mu)\) is the set of all functions in \(\mathcal{M}(X,\mathcal{A})\) which are essentially \(\mu\)-bounded over \(X\) and \(L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}\). It is established that an ideal \(I\) in \(\mathcal{M}(X,\mathcal{A})\) is dense in the \(U_\mu\)-topology if and only if it is dense in the \(m_\mu\)-topology and this happens when and only when there exists \(Z\in Z[I]\) such that \(\mu(Z)=0\). Furthermore, it is proved that \(I\) is closed in \(\mathcal{M}(X,\mathcal{A})\) in the \(m_\mu\)-topology if and only if it is a \(Z_\mu\)-ideal in the sense that if \(f\equiv g\) almost everywhere on \(X\) with \(f\in I\) and \(g\in\mathcal{M}(X,\mathcal{A})\), then \(g\in I\).