A Generalization of $m$-topology and $U$-topology on rings of measurable functions

Abstract

For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $\mu$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called $m_{\mu }$ and $U_{\mu }$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be referred to as the $m_{{\mu }_I}$ and $U_{{\mu }_I}$ topology, respectively, throughout the paper. $L_I^{\infty }\left(\mu \right)$ stands for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are essentially $I$-bounded (over the measure space ($X,\mathcal{A},\mu$)). Also let $I_{\mu }(X,\mathcal{A})=\{f\in \mathcal{M}(X,\mathcal{A}):\text{for every}g\in \mathcal{M}(X,\mathcal{A}),fg\text{is essentially}I$-$\text{bounded}\}$. Then $I_{\mu }(X,\mathcal{A})$ is an ideal in $\mathcal{M}(X,\mathcal{A})$ containing $I$ and contained in $L_I^{\infty }\left(\mu \right)$. It is also shown that $I_{\mu }(X,\mathcal{A})$ and $L_I^{\infty }\left(\mu \right)$ are the components of $0$ in the spaces $m_{{\mu }_I}$ and $U_{{\mu }_I}$, respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide.