Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914759032438784 |
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| author | Sobota, Damian Zdomskyy, Lyubomyr |
| author_facet | Sobota, Damian Zdomskyy, Lyubomyr |
| contents | Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a Grothendieck space but there exists a sequence $(μ_n)$ of Radon measures on $K$ such that $μ_n(A)\to0$ for every clopen set $A\subseteq K$ and $\int_Kfdμ_n\not\to0$ for some $f\in C(K)$. Consequently, we get that Martin's axiom implies the existence of a Boolean algebra with the Grothendieck property but without the Nikodym property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16155 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property Sobota, Damian Zdomskyy, Lyubomyr Functional Analysis Logic Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a Grothendieck space but there exists a sequence $(μ_n)$ of Radon measures on $K$ such that $μ_n(A)\to0$ for every clopen set $A\subseteq K$ and $\int_Kfdμ_n\not\to0$ for some $f\in C(K)$. Consequently, we get that Martin's axiom implies the existence of a Boolean algebra with the Grothendieck property but without the Nikodym property. |
| title | Construction under Martin's axiom of a Boolean algebra with the Grothendieck property but without the Nikodym property |
| topic | Functional Analysis Logic |
| url | https://arxiv.org/abs/2312.16155 |