Kuratowski–Ulam theorem


In mathematics, the Kuratowski–Ulam theorem, introduced by, called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second countable Baire spaces.
Let X and Y be second countable Baire spaces, and let. Then the following are equivalent if A has the Baire property:
  1. A is meager.
  2. The set is comeager in X, where, where is the projection onto Y.
Even if A does not have the Baire property, 2. follows from 1.
Note that the theorem still holds for X an arbitrary Hausdorff space and Y a Hausdorff space with countable π-base.
The theorem is analogous to the regular Fubini's theorem for the case where the considered function is a characteristic function of a subset in a product space, with the usual correspondences, namely, meagre set with a set of measure zero, comeagre set with one of full measure, and a set with the Baire property with a measurable set.