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Patar knight (talk | contribs) Adding local short description: "Measure theory and probability theorem", overriding Wikidata description "theorem" |
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{{Short description|Measure theory and probability theorem}}
In [[Measure (mathematics)|measure theory]] and [[Probability theory|probability]], the '''monotone class theorem''' connects monotone classes and [[Sigma-algebra|{{sigma}}-algebra]]s. The theorem says that the smallest [[#Definition of a monotone class|monotone class]] containing an [[Field of sets|algebra of sets]] <math>G</math> is precisely the smallest [[Sigma-algebra|{{sigma}}-algebra]] containing <math>G.</math> It is used as a type of [[transfinite induction]] to prove many other theorems, such as [[Fubini's theorem]].
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==Results and applications==
As a corollary, if <math>G</math> is a [[ring of sets]]
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a [[Sigma-algebra|{{sigma}}-algebra]].
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