Monotone class theorem
{{Short description|Measure theory and probability theorem}}
In measure theory and probability, the monotone class theorem connects monotone classes and {{sigma}}-algebras. The theorem says that the smallest monotone class containing an algebra of sets is precisely the smallest {{sigma}}-algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
Definition of a monotone class
Monotone class theorem for sets
{{math theorem|name=Monotone class theorem for sets|note=|style=|math_statement=
Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the {{sigma}}-algebra generated by ; that is
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Monotone class theorem for functions
{{math theorem|name=Monotone class theorem for functions|note=|style=|math_statement=
Let be a {{pi}}-system that contains and let be a collection of functions from to with the following properties:
- If then where denotes the indicator function of
- 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
}}
=Proof=
The following argument originates in Rick Durrett's Probability: Theory and Examples.{{cite book|last=Durrett|first=Rick|year=2010|title=Probability: Theory and Examples|url=https://archive.org/details/probabilitytheor00rdur|url-access=limited|edition=4th|publisher=Cambridge University Press|page=[https://archive.org/details/probabilitytheor00rdur/page/n287 276]|isbn=978-0521765398}}
{{math proof|drop=hidden|proof=
The assumption (2), and (3) imply that is a {{lambda}}-system.
By (1) and the {{pi}}−{{lambda}} theorem,
Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to
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Results and applications
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the {{sigma}}-ring of
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a {{sigma}}-algebra.
The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.
See also
- {{annotated link|Dynkin system}}
- {{annotated link|π-λ theorem|{{pi}}-{{lambda}} theorem}}
- {{annotated link|Pi-system|{{pi}}-system}}
- {{annotated link|σ-algebra}}
Citations
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References
- {{Durrett Probability Theory and Examples 5th Edition}}