Measure finite topology on the ring of measurable functions

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}.