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

TL;DR

This paper introduces a generalized framework for $m$-topology and $U$-topology on rings of measurable functions using ideals, analyzing their properties and conditions for equivalence.

Contribution

It extends existing topologies on measurable function rings by incorporating ideals, providing new insights into their structure and relationships.

Findings

Defined generalized $m_{ ext{measure}}$ and $U_{ ext{measure}}$ topologies using ideals.

Characterized the ideal $I_{ ext{measure}}$ and subring $L_{I}^{ ext{infty}}( ext{measure})$.

Established conditions under which the generalized topologies coincide.

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.