Kuratowski–Ulam theorem

Template:Short description In mathematics, the Kuratowski–Ulam theorem, introduced by Template:Harvs, 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:

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 $A_x={\pi }_Y[A\cap \{x\}\times Y]$, 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 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

• Template:Cite journal

Template:Topology-stub