{"title":"具有不可分割度量的可数紧密对偶球","authors":"Piotr Koszmider, Zdeněk Silber","doi":"10.1112/jlms.12988","DOIUrl":null,"url":null,"abstract":"<p>We construct a compact Hausdorff space <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> such that the space <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K)$</annotation>\n </semantics></math> of Radon probability measures on <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> considered with the <span></span><math>\n <semantics>\n <msup>\n <mtext>weak</mtext>\n <mo>∗</mo>\n </msup>\n <annotation>$\\text{weak}^*$</annotation>\n </semantics></math> topology (induced from the space of continuous functions <span></span><math>\n <semantics>\n <mrow>\n <mi>C</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$C(K)$</annotation>\n </semantics></math>) is countably tight that is a generalization of sequentiality (i.e., if a measure <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math> is in the closure of a set <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$M$</annotation>\n </semantics></math>, there is a countable <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>M</mi>\n <mo>′</mo>\n </msup>\n <mo>⊆</mo>\n <mi>M</mi>\n </mrow>\n <annotation>$M^{\\prime }\\subseteq M$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math> is in the closure of <span></span><math>\n <semantics>\n <msup>\n <mi>M</mi>\n <mo>′</mo>\n </msup>\n <annotation>$M^{\\prime }$</annotation>\n </semantics></math>) but <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> carries a Radon probability measure that has uncountable Maharam type (i.e., <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>L</mi>\n <mn>1</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>μ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$L_1(\\mu)$</annotation>\n </semantics></math> is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the <span></span><math>\n <semantics>\n <mo>◇</mo>\n <annotation>$\\diamondsuit$</annotation>\n </semantics></math> principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>×</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K\\times K)$</annotation>\n </semantics></math> implies that all Radon measures on <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> have countable type. So, our example shows that the tightness of <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>×</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K\\times K)$</annotation>\n </semantics></math> and of <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n <mo>×</mo>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K)\\times P(K)$</annotation>\n </semantics></math> can be different as well as <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K)$</annotation>\n </semantics></math> may have Corson property (C), while <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>×</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$P(K\\times K)$</annotation>\n </semantics></math> fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez, and Rueda Zoca.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Countably tight dual ball with a nonseparable measure\",\"authors\":\"Piotr Koszmider, Zdeněk Silber\",\"doi\":\"10.1112/jlms.12988\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We construct a compact Hausdorff space <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> such that the space <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K)$</annotation>\\n </semantics></math> of Radon probability measures on <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> considered with the <span></span><math>\\n <semantics>\\n <msup>\\n <mtext>weak</mtext>\\n <mo>∗</mo>\\n </msup>\\n <annotation>$\\\\text{weak}^*$</annotation>\\n </semantics></math> topology (induced from the space of continuous functions <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>C</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$C(K)$</annotation>\\n </semantics></math>) is countably tight that is a generalization of sequentiality (i.e., if a measure <span></span><math>\\n <semantics>\\n <mi>μ</mi>\\n <annotation>$\\\\mu$</annotation>\\n </semantics></math> is in the closure of a set <span></span><math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$M$</annotation>\\n </semantics></math>, there is a countable <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>M</mi>\\n <mo>′</mo>\\n </msup>\\n <mo>⊆</mo>\\n <mi>M</mi>\\n </mrow>\\n <annotation>$M^{\\\\prime }\\\\subseteq M$</annotation>\\n </semantics></math> such that <span></span><math>\\n <semantics>\\n <mi>μ</mi>\\n <annotation>$\\\\mu$</annotation>\\n </semantics></math> is in the closure of <span></span><math>\\n <semantics>\\n <msup>\\n <mi>M</mi>\\n <mo>′</mo>\\n </msup>\\n <annotation>$M^{\\\\prime }$</annotation>\\n </semantics></math>) but <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> carries a Radon probability measure that has uncountable Maharam type (i.e., <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>L</mi>\\n <mn>1</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>μ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$L_1(\\\\mu)$</annotation>\\n </semantics></math> is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the <span></span><math>\\n <semantics>\\n <mo>◇</mo>\\n <annotation>$\\\\diamondsuit$</annotation>\\n </semantics></math> principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>×</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K\\\\times K)$</annotation>\\n </semantics></math> implies that all Radon measures on <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> have countable type. So, our example shows that the tightness of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>×</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K\\\\times K)$</annotation>\\n </semantics></math> and of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n <mo>×</mo>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K)\\\\times P(K)$</annotation>\\n </semantics></math> can be different as well as <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K)$</annotation>\\n </semantics></math> may have Corson property (C), while <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>×</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$P(K\\\\times K)$</annotation>\\n </semantics></math> fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez, and Rueda Zoca.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12988\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12988","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
We construct a compact Hausdorff space K $K$ such that the space P ( K ) $P(K)$ of Radon probability measures on K $K$ considered with the weak ∗ $\text{weak}^*$ topology (induced from the space of continuous functions C ( K ) $C(K)$ ) is countably tight that is a generalization of sequentiality (i.e., if a measure μ $\mu$ is in the closure of a set M $M$ , there is a countable M ′ ⊆ M $M^{\prime }\subseteq M$ such that μ $\mu$ is in the closure of M ′ $M^{\prime }$ ) but K $K$ carries a Radon probability measure that has uncountable Maharam type (i.e., L 1 ( μ ) $L_1(\mu)$ is nonseparable).这个构造(必然)使用了一个额外的集合论假设(◇ $\diamondsuit$ 原则),因为根据弗雷姆林的一个结果,我们已经知道这样的空间是不存在的。这应该与普莱巴内克和索博塔的结果相比较,他们证明了 P ( K × K ) $P(K\times K)$ 的可数紧密性意味着 K $K$ 上的所有拉顿量都具有可数类型。因此,我们的例子表明,P ( K × K ) $P(K\times K)$ 和 P ( K ) × P ( K ) $P(K)\times P(K)$ 的紧密性可能不同,P ( K ) $P(K)$ 可能具有 Corson 性质 (C),而 P ( K × K ) $P(K\times K)$ 则不具有,这回答了一个 Pol 问题。我们的构造也是巴拿赫空间注入张量积一般背景下的一个相关例子,补充了阿维莱斯、马丁内斯-塞万提斯、罗德里格斯和鲁埃达-佐卡的最新成果。
Countably tight dual ball with a nonseparable measure
We construct a compact Hausdorff space such that the space of Radon probability measures on considered with the topology (induced from the space of continuous functions ) is countably tight that is a generalization of sequentiality (i.e., if a measure is in the closure of a set , there is a countable such that is in the closure of ) but carries a Radon probability measure that has uncountable Maharam type (i.e., is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of implies that all Radon measures on have countable type. So, our example shows that the tightness of and of can be different as well as may have Corson property (C), while fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avilés, Martínez-Cervantes, Rodríguez, and Rueda Zoca.
期刊介绍:
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