Measure finite topology on the ring of measurable functions

Abstract

Let $\mathcal{M}(X,\mathcal{A},\mu )$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},\mu )$. A topology on $\mathcal{M}(X,\mathcal{A},\mu )$, called the {$F_{\mu }$-topology} weaker than the { $U_{\mu }$-topology} is introduced. It is realized that the {component}, the {quasi component} and the {path component }in this {$F_{\mu }$-topology} are identical. It turns out that the {$F_{\mu }$-topology} on $\mathcal{M}(X,\mathcal{A},\mu )$ becomes {connected} if and only if it is {path connected} if and only if $\mu$ is an {atomic measure} of a special type. It is also proved that the {$F_{\mu }$-topology} is {first countable} when and only when $\mu$ is a {hemifinite measure.} Finally, it is shown that the {second countability} of the {$F_{\mu }$-topology} is equivalent to the {hemifiniteness} of the measure $\mu$ together with the {countable chain condition} of the {$F_{\mu }$-topology}.