Kuratowski–Ulam theorem explained

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 x Y . Then the following are equivalent if A has the Baire property:

1. A is meager (respectively comeager).

\{ x \in X :A _{x} ismeager(resp.comeager)in Y \} is comeager in X, where A _{x=\pi} _{Y[A\cap} \lbrace x \rbrace x Y] , where \pi _{Y} is the projection onto Y.Even if A does not have the Baire property, 2. follows from 1.^[1] Note that the theorem still holds (perhaps vacuously) 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.

References

• Quelques propriétés topologiques du produit combinatoire. Kazimierz Kuratowski. Kazimierz . Kuratowski . Stanislaw Ulam. Stanislaw. Ulam . Institute of Mathematics Polish Academy of Sciences. Fundamenta Mathematicae. 1932. 19. 1. 247–251. 10.4064/fm-19-1-247-251. free.

Notes and References

1. Book: Srivastava, Shashi Mohan . [{{Google books |plainurl=yes |id=FhYGYJtMwcUC |page=112}} A Course on Borel Sets ]. Graduate Texts in Mathematics . Berlin . Springer . 1998 . 180 . 0-387-98412-7. 10.1007/978-3-642-85473-6 . 1619545 . 112 .