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