In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam ( 1932), 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 ${\displaystyle A\subset X\times Y}$. Then the following are equivalent if A has the Baire property:

1. A is meager (respectively comeager).
2. The set ${\displaystyle \{x\in X:A_{x}{\text{ is meager (resp. comeager) in }}Y\}}$ is comeager in X, where ${\displaystyle A_{x}=\pi _{Y}[A\cap \lbrace x\rbrace \times Y]}$, where ${\displaystyle \pi _{Y}}$ 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

• Kuratowski, Kazimierz; Ulam, Stanislaw (1932). "Quelques propriétés topologiques du produit combinatoire" (PDF). Fundamenta Mathematicae. 19 (1). Institute of Mathematics Polish Academy of Sciences: 247–251. doi: 10.4064/fm-19-1-247-251.

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