M-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS

Abstract

In this article we consider the $m$-topology on linebreak $M(X,mathscr{A})$, the ring of all real measurable functions on a measurable space $(X, mathscr{A})$, and we denote it by $M_m(X,mathscr{A})$. We show that $M_m(X,mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,mathscr{A})‎cong‎ mathbb R^n$ as topological rings. Also, we show that $M_m(X,mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,mathscr{A})= {M}_{u}(X,mathscr{A})$, if and only if $ M_m(X,mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,mathscr{A})$ is a connected space, if and only if $M_m(X,mathscr{A})$ is a locally connected space, if and only if $M^*(X,mathscr{A})$ is a connected subset of $M_m(X,mathscr{A})$.