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}.
Categories: math.GN math.AC