On maldistributed sequences and meager ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916760287969280 |
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| author | Leonetti, Paolo |
| author_facet | Leonetti, Paolo |
| contents | We show that an ideal $\mathcal{I}$ on $ω$ is meager if and only if the set of sequences $(x_n)$ taking values in a Polish space $X$ for which all elements of $X$ are $\mathcal{I}$-cluster points of $(x_n)$ is comeager. The latter condition is also known as $ν$-maldistribution, where $ν: \mathcal{P}(ω)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $ν(A)=1$ if and only if $A\notin \mathcal{I}$. It turns out that the meagerness of $\mathcal{I}$ is also equivalent to a technical condition given by Misik and Toth in [J. Math. Anal. Appl. 541 (2025), 128667]. Lastly, we show that the analogue of the first part holds replacing $ν$ with $\|\cdot\|_φ$, where $φ$ is a lower semicontinuous submeasure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On maldistributed sequences and meager ideals Leonetti, Paolo General Topology Functional Analysis We show that an ideal $\mathcal{I}$ on $ω$ is meager if and only if the set of sequences $(x_n)$ taking values in a Polish space $X$ for which all elements of $X$ are $\mathcal{I}$-cluster points of $(x_n)$ is comeager. The latter condition is also known as $ν$-maldistribution, where $ν: \mathcal{P}(ω)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $ν(A)=1$ if and only if $A\notin \mathcal{I}$. It turns out that the meagerness of $\mathcal{I}$ is also equivalent to a technical condition given by Misik and Toth in [J. Math. Anal. Appl. 541 (2025), 128667]. Lastly, we show that the analogue of the first part holds replacing $ν$ with $\|\cdot\|_φ$, where $φ$ is a lower semicontinuous submeasure. |
| title | On maldistributed sequences and meager ideals |
| topic | General Topology Functional Analysis |
| url | https://arxiv.org/abs/2505.20490 |