Kuratowski–Ulam theorem: Difference between revisions
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In mathematics, the '''Kuratowski–Ulam theorem''', introduced by {{harvs|txt|author1-link= Kazimierz Kuratowski|first1=Kazimierz |last=Kuratowski |author2-link=Stanislaw Ulam|first2=Stanislaw|last2= Ulam|year=1932}}, called also Fubini theorem for category, is an analog of the [[Fubini's theorem]] for arbitrary [[second countable]] [[Baire space]]s. |
In mathematics, the '''Kuratowski–Ulam theorem''', introduced by {{harvs|txt|author1-link= Kazimierz Kuratowski|first1=Kazimierz |last=Kuratowski |author2-link=Stanislaw Ulam|first2=Stanislaw|last2= Ulam|year=1932}}, called also Fubini theorem for category, is an analog of the [[Fubini's theorem]] for arbitrary [[second countable]] [[Baire space]]s. |
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Let ''X'' and ''Y'' be second countable Baire spaces (or, in particular, [[Polish space]]s), and <math>A\subset X\times Y</math>. Then the following are equivalent if ''A'' has the [[Property of Baire|Baire property]]: |
Let ''X'' and ''Y'' be second countable Baire spaces (or, in particular, [[Polish space]]s), and <math>A\subset X\times Y</math>. Then the following are equivalent if ''A'' has the [[Property of Baire|Baire property]]: |
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# ''A'' is [[Meager set|meager]] (respectively comeager) |
# ''A'' is [[Meager set|meager]] (respectively comeager). |
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# The set <math>\{ x \in X :A_x \text{ is meager (resp. comeager) in }Y \}</math> is comeager in X, where <math>A_x=\pi_Y[A\cap \lbrace x \rbrace \times Y]</math>, where <math>\pi_Y</math> is the projection onto Y. |
# The set <math>\{ x \in X :A_x \text{ is meager (resp. comeager) in }Y \}</math> is comeager in X, where <math>A_x=\pi_Y[A\cap \lbrace x \rbrace \times Y]</math>, where <math>\pi_Y</math> is the projection onto ''Y''. |
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Even if A does not have the Baire property, 2. follows from 1.<ref>{{cite book |first= |
Even if ''A'' does not have the Baire property, 2. follows from 1.<ref>{{cite book |first=Shashi Mohan |last=Srivastava |title=A Course on Borel Sets |location=Berlin |publisher=Springer |year=1998 |isbn=0-387-98412-7|doi=10.1007/978-3-642-85473-6 |mr=1619545 |url={{Google books |plainurl=yes |id=FhYGYJtMwcUC |page=112}} |page=112 }}</ref> |
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Note that the theorem still holds (perhaps vacuously) for X |
Note that the theorem still holds (perhaps vacuously) for ''X'' an arbitrary Hausdorff space and ''Y'' a Hausdorff space with countable [[π-base]]. |
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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. |
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. |
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==References== |
==References== |
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{{Reflist}} |
{{Reflist}} |
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*{{cite journal |
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*{{citation |
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|title=Quelques propriétés topologiques du produit combinatoire |
|title=Quelques propriétés topologiques du produit combinatoire |
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|author1-link= Kazimierz Kuratowski|first1=Kazimierz |last=Kuratowski |
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|first=C.|last= Kuratowski |first2= St.|last2= Ulam |
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|author2-link=Stanislaw Ulam|first2=Stanislaw|last2= Ulam| |
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|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm19/fm19121.pdf |
|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm19/fm19121.pdf |
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|publisher= Institute of Mathematics Polish Academy of Sciences |
|publisher= Institute of Mathematics Polish Academy of Sciences |
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|journal= Fundamenta Mathematicae |
|journal= [[Fundamenta Mathematicae]] |
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|year= 1932 |
|year= 1932 |
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|volume= 19 |
|volume= 19 |
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Revision as of 21:17, 21 March 2020
In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam (1932), called also Fubini theorem for category, 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.
![{\displaystyle A_{x}=\pi _{Y}[A\cap \lbrace x\rbrace \times Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/078a78f4df5ddab6a60a933da44a60353bfd7a57)
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 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, Shashi Mohan (1998). A Course on Borel Sets. Berlin: Springer. p. 112. doi:10.1007/978-3-642-85473-6. ISBN 0-387-98412-7. MR 1619545.
- 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.
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