GAMES AND HEREDITARY BAIRENESS IN HYPERSPACES AND SPACES OF PROBABILITY MEASURES

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 F and prove that it is equivalent to K(F) being hereditarily Baire. We also show that if X is separable metrizable and K(X) is hereditarily Baire, then the space Pr(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 Pr(X) hereditarily Baire. As far as we know, this is the first example of this kind.