Countably tight dual ball with a nonseparable measure

IF 1 2区 数学 Q1 MATHEMATICS
Piotr Koszmider, Zdeněk Silber
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引用次数: 0

Abstract

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). The construction uses (necessarily) an additional set-theoretic assumption (the $\diamondsuit$ 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 P ( K × K ) $P(K\times K)$ implies that all Radon measures on K $K$ have countable type. So, our example shows that the tightness of P ( K × K ) $P(K\times K)$ and of P ( K ) × P ( K ) $P(K)\times P(K)$ can be different as well as P ( K ) $P(K)$ may have Corson property (C), while P ( K × K ) $P(K\times K)$ 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.

具有不可分割度量的可数紧密对偶球
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。
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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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