Kuratowski–Ulam theorem

In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also Fubini theorem for category, is an analog of the Fubini's theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces), and $A\subset X\times Y$. Then the following are equivalent if A has the Baire property:

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.

Even if A does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for X - arbitrary Hausdorff space and Y - Hausdorff with countable π-base.

The theorem is analogous to regular Fubini's theorem for the case where the considered function is a characteristic function of a set in a product space, with usual correspondences – meagre set with set of measure zero, comeagre set with one of full measure, a set with Baire property with a measurable set.

References

• Kuratowski, C.; Ulam, St. (1932), "Quelques propriétés topologiques du produit combinatoire" (PDF), Fundamenta Mathematicae, 19 (1), Institute of Mathematics Polish Academy of Sciences: 247–251