  It is well known that if the nonempty player of the Banach-Mazur game has a
winning strategy on a space, then that space is Baire in all powers even in the
box topology. The converse of this implication may be true also: We know of no
consistency result to the contrary. In this paper we establish the consistency
of the converse relative to the consistency of the existence of a proper class
of measurable cardinals.
