$U$-topology and $m$-topology on the ring of Measurable Functions, generalized and revisited

Abstract

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 $g\in I$.