论文标题
在超空间和概率措施的空间中的游戏和遗传性baireneses
Games and hereditary Baireness in hyperspaces and spaces of probability measures
论文作者
论文摘要
我们确定在某些拓扑游戏中存在获胜策略，与强大的Choquet游戏密切相关，在拓扑空间$ x $及其Hyperspace $ k（x）$中，所有非空的紧凑型子集的$ x $，配备了越野鞋的拓扑，是玩家中的一位。对于可分离的Metrizable Space $ x $，我们确定了相当于$ k（x）$的游戏理论状况。这很容易意味着Gartside，Medini和Zdomskyy的最新结果，这些结果是超大型Metrizable Spaces $ x $ $ k（x）$ $ x $的遗传性baire属性的特征。随后，我们使用拓扑游戏在概率措施和超过自然数量的过滤器中研究遗传性baire属性。为此，我们引入了一个强$ p $ -filter $ \ Mathcal {f} $的概念，并证明它等于$ k（\ nathcal {f}）$是遗传性的。我们还表明，如果$ x $是可分离的，而$ k（x）$是遗传性的，那么$ x $上的borel borel概率ra（x）$的空间$ p_r（x）$也是herditit上的贝尔。因此，存在（在ZFC中）可分离的Metrizable Space $ x $，与$ p_r（x）$ herdityly baire无法完全METRIALIAL。据我们所知，这是此类的第一个例子。
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.
