A pseudometric on $\mathcal{M}(X,\mathscr{A})$ induced by a measure
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915305403449344 |
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| author | Dey, Amrita |
| author_facet | Dey, Amrita |
| contents | For a probability measure space $(X,\mathscr{A},μ)$, we define a pseudometric $δ$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $δ(f,g)=μ(X\setminus Z(f-g))$ and denote the topological space induced by $δ$ as $\mathcal{M}_δ$. We examine several topological properties, such as connectedness, compactness, Lindelöfness, separability and second countability of this pseudometric space. We realise that the space is connected if and only if $μ$ is a non-atomic measure and we explicitly describe the components in $\mathcal{M}_δ$, for any choice of measure. We also deduce that $\mathcal{M}_δ$ is zero-dimensional if and only if $μ$ is purely atomic. We define $μ$ to be bounded away from zero, if every non-zero measurable set has measure greater than some constant. We establish several conditions equivalent to $μ$ being bounded away from zero. For instance, $μ$ is bounded away from zero if and only if $\mathcal{M}_δ$ is a locally compact space. We conclude this article by describing the structure of compact sets and Lindelöf sets in $\mathcal{M}_δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A pseudometric on $\mathcal{M}(X,\mathscr{A})$ induced by a measure Dey, Amrita General Topology 54C35, 28A10, 54E35 For a probability measure space $(X,\mathscr{A},μ)$, we define a pseudometric $δ$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $δ(f,g)=μ(X\setminus Z(f-g))$ and denote the topological space induced by $δ$ as $\mathcal{M}_δ$. We examine several topological properties, such as connectedness, compactness, Lindelöfness, separability and second countability of this pseudometric space. We realise that the space is connected if and only if $μ$ is a non-atomic measure and we explicitly describe the components in $\mathcal{M}_δ$, for any choice of measure. We also deduce that $\mathcal{M}_δ$ is zero-dimensional if and only if $μ$ is purely atomic. We define $μ$ to be bounded away from zero, if every non-zero measurable set has measure greater than some constant. We establish several conditions equivalent to $μ$ being bounded away from zero. For instance, $μ$ is bounded away from zero if and only if $\mathcal{M}_δ$ is a locally compact space. We conclude this article by describing the structure of compact sets and Lindelöf sets in $\mathcal{M}_δ$. |
| title | A pseudometric on $\mathcal{M}(X,\mathscr{A})$ induced by a measure |
| topic | General Topology 54C35, 28A10, 54E35 |
| url | https://arxiv.org/abs/2505.19780 |