| dbo:abstract |
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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dbpedia-fr:Lemme_de_classe_monotone |
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dbr:Ring_of_sets dbr:Transfinite_induction dbr:Measure_(mathematics) dbc:Theorems_in_measure_theory dbr:Sigma-algebra dbr:Closure_(mathematics) dbr:Fubini's_theorem dbr:Dynkin_system dbr:Field_of_sets dbr:Family_of_sets dbr:Rick_Durrett dbr:Probability_theory dbc:Families_of_sets dbr:Pi_system |
| dbp:drop |
hidden (en) |
| dbp:mathStatement |
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) |
| dbp:name |
Monotone class theorem for functions (en) Monotone class theorem for sets (en) |
| dbp:proof |
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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dbc:Theorems_in_measure_theory dbc:Families_of_sets |
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yago:WikicatMathematicalTheorems yago:WikicatTheoremsInMeasureTheory yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293 |
| rdfs:comment |
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) |
| rdfs:label |
Satz über monotone Klassen (de) Monotone class theorem (en) Теорема про монотонний клас (uk) |
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freebase:Monotone class theorem yago-res:Monotone class theorem wikidata:Monotone class theorem dbpedia-de:Monotone class theorem dbpedia-he:Monotone class theorem dbpedia-uk:Monotone class theorem https://global.dbpedia.org/id/2H2fq |
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wikipedia-en:Monotone_class_theorem?oldid=1106927279&ns=0 |
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