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.
A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to linear dynamics
