# Kuratowski–Ulam theorem

In mathematics, the **Kuratowski–Ulam theorem**, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also Fubini theorem for categories, 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
. Then the following are equivalent if *A* has the Baire property:

*A*is meager (respectively comeager)- 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.^{[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

- ↑ Srivastava, S. (1998).
*A Course on Borel Sets*. Berlin: Springer. p. 112. ISBN 0-387-98412-7.

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