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.

中文翻译：

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超空间和概率测度空间中的博弈和遗传

我们确定在某些拓扑博弈中存在获胜策略，与强 Choquet 博弈密切相关，在拓扑空间中进行 $X$ 及其超空间 $K(X)$ 的所有非空紧子集 $X$ 配备了 Vietoris 拓扑，相当于玩家之一。对于可分离的可计量空间 $X$ ，我们确定一个博弈论条件等价于 $K(X)$ 是世袭的Baire。它很容易暗示了 Gartside、Medini 和 Zdomskyy 最近的一个结果，该结果表征了超空间的遗传性 Baire 属性 $K(X)$ 在可分离的可计量空间上 $X$ 通过紧化的余数的门格尔性质 $X$ . 随后，我们使用拓扑博弈来研究概率测度空间和自然数过滤器上的超空间中的遗传 Baire 属性。为此，我们引入了强 $P$ -筛选 ${\mathcal{F}}$ 并证明它等价于 $K({\mathcal{F}})$ 是世袭的Baire。我们还表明，如果 $X$ 是可分离的，可度量的并且 $K(X)$ 遗传上是Baire，那么空间 $P_{r}(X)$ Borel 概率氡测量 $X$ 也是世袭的Baire。由此可见（在 ZFC 中）存在一个可分离的可度量空间 $X$ , 这不是完全可度量的 $P_{r}(X)$ 世袭拜尔。据我们所知，这是此类的第一个例子。