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 $I_{\mu} (X,\mathcal{A}) = \big \{ f \in \mathcal{M}(X,\mathcal{A}) : \, \text{for every} \, g \in \mathcal{M}(X,\mathcal{A}), fg \, \, \text{is essentially} \, I$-$\text{bounded} \big \}$. 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.