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- Der Satz über monotone Klassen ist ein zentraler Satz der Maßtheorie, dem Teilgebiet der Mathematik, das sich mit den Eigenschaften von Maßräumen und Funktionen auf ihnen beschäftigt. (de)
- In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest containing an algebra of sets is precisely the smallest 𝜎-algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem. (en)
- Теорема про монотонний клас — твердження у теорії міри і теорії ймовірностей про рівність монотонного класу і σ-кільця породжених деяким кільцем множин. (uk)
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- 4518 (xsd:nonNegativeInteger)
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- Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the -algebra generated by ; that is, (en)
- Let be a -system that contains and let be a collection of functions from to with the following properties:
# If then
# If and then and
# If is a sequence of non-negative functions that increase to a bounded function then
Then contains all bounded functions that are measurable with respect to which is the sigma-algebra generated by (en)
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- Monotone class theorem for functions (en)
- Monotone class theorem for sets (en)
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- The assumption , and imply that is a -system.
By and the − theorem,
Statement implies that contains all simple functions, and then implies that contains all bounded functions measurable with respect to (en)
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- Der Satz über monotone Klassen ist ein zentraler Satz der Maßtheorie, dem Teilgebiet der Mathematik, das sich mit den Eigenschaften von Maßräumen und Funktionen auf ihnen beschäftigt. (de)
- In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest containing an algebra of sets is precisely the smallest 𝜎-algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem. (en)
- Теорема про монотонний клас — твердження у теорії міри і теорії ймовірностей про рівність монотонного класу і σ-кільця породжених деяким кільцем множин. (uk)
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- Satz über monotone Klassen (de)
- Monotone class theorem (en)
- Теорема про монотонний клас (uk)
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