Fubini's Theorem - Kuratowski-Ulam Theorem

Kuratowski-Ulam Theorem

The Kuratowski-Ulam theorem, named after Polish mathematicians Kazimierz Kuratowski and Stanisław Ulam, called also Fubini theorem for category, is a similar result 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:

1. A is meager (respectively comeager)
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 (perhaps vacuously) for X - arbitrary Hausdorff space and Y - Hausdorff with countable π-base.

The theorem is analogous to regular Fubini 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.