u-topology and m-topology on the ring of measurable functions, generalized and revisited

Abstract

Given an ideal I in the ring \(\mathcal {M}(X,\mathcal {A})\) of all real valued measurable functions over the measurable space \((X,\mathcal {A})\) and a measure \(\mu :\mathcal {A}\rightarrow [0,\infty ]\), we introduce the \(u_\mu ^I\)-topology and the \(m_\mu ^I\)-topology on \(\mathcal {M}(X,\mathcal {A})\) as generalizations of the u-topology and the m-topology on \(\mathcal {M}(X,\mathcal {A})\) respectively. For a countably generated ideal I in \(\mathcal {M}(X,\mathcal {A})\), it is proved that 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 both of these topologies are determined and it is proved that the condition of denseness of an ideal I in \(\mathcal {M}(X,\mathcal {A})\) is equivalent in these two topologies and this happens when and only when there exists \(Z\in Z[I]\) such that \(\mu (Z)=0\). It is also 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. Two more topologies on \(\mathcal {M}(X,\mathcal {A})\) viz. the \(u_{\mu ,F}^I\)-topology and the \(m_{\mu ,F}^I\)-topology, finer than the \(u_\mu ^I\)-topology and the \(m_\mu ^I\)-topology respectively are introduced and a few relevant properties are investigated thereon.