Measure finite topology on the ring of measurable functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917092619452416 |
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| author | Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy |
| author_facet | Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure finite topology on the ring of measurable functions Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy General Topology Commutative Algebra 54C40, 46E30 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}. |
| title | Measure finite topology on the ring of measurable functions |
| topic | General Topology Commutative Algebra 54C40, 46E30 |
| url | https://arxiv.org/abs/2511.15436 |