Monotone class theorem: Difference between revisions

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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&nbsp;<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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{{math theorem|name=Monotone class theorem for sets|note=|style=|math_statement=

Let <math>G</math> be an [[Field of sets|algebra of sets]] and define <math>M(G)</math> to be the smallest monotone class containing <math>G.</math> Then <math>M(G)</math> is precisely the [[Sigma-algebra|{{sigma}}-algebra]] generated by <math>G;</math>; that is <math>\sigma(G) = M(G).</math>

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==Results and applications==

As a corollary, if <math>G</math> is a [[ring of sets]] of sets, then the smallest monotone class containing it coincides with the [[Sigma-ring|{{sigma}}-ring]] of <math>G.</math>

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a [[Sigma-algebra|{{sigma}}-algebra]].