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 (or, in particular, Polish spaces), and let $$A \subset X \times Y$$. Then the following are equivalent if A has the Baire property: Even if A does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for X an arbitrary Hausdorff space and Y a Hausdorff space with countable π-base.
 * 1) A is meager (respectively comeager).
 * 2) The set $$\{ x \in X :A_x \text{ is meager (resp. comeager) in }Y \}$$ is comeager in X, where $$A_x=\pi_Y[A\cap \lbrace x \rbrace \times Y]$$, where $$\pi_Y$$ is the projection onto Y.

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.