超空间与概率测度空间中的博弈与遗传Baire性质
Games and hereditary Baireness in hyperspaces and spaces of probability measures
AI总结：
本文用拓扑博弈刻画超空间的遗传Baire性质，并推广到概率测度空间与滤子超空间，给出ZFC中首个非完全可度量化但$P_r(X)$遗传Baire的例子。
摘要：
我们证明，在拓扑空间$X$及其超空间$K(X)$（即$X$的所有非空紧子集构成的空间，赋予Vietoris拓扑）中进行的、与强Choquet博弈密切相关的某些拓扑博弈里，某一位参与者存在获胜策略，这两件事是等价的。对于可分可度量化空间$X$，我们识别出一个博弈论条件，它等价于$K(X)$具有遗传Baire性质。该条件能相当容易地推出Gartside、Medini和Zdomskyy最近的一项结果：他们通过$X$的某个紧化之余集的Menger性质，刻画了可分可度量化空间$X$上的超空间$K(X)$的遗传Baire性质。随后，我们利用拓扑博弈研究概率测度空间以及自然数上滤子所对应超空间中的遗传Baire性质。为此，我们引入强$P$-滤子$\mathcal{F}$的概念，并证明它等价于$K(\mathcal{F})$具有遗传Baire性质。我们还表明，如果$X$可分可度量化且$K(X)$具有遗传Baire性质，那么$X$上Borel概率Radon测度构成的空间$P_r(X)$也具有遗传Baire性质。由此可知，在ZFC中存在一个并非完全可度量化的可分可度量化空间$X$，但$P_r(X)$具有遗传Baire性质。据我们所知，这是此类例子中的首个。
英文摘要：
We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter $\mathcal{F}$ and prove that it is equivalent to $K(\mathcal{F})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_r(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ which is not completely metrizable with $P_r(X)$ hereditarily Baire. As far as we know this is the first example of this kind.
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