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Monotone class theorem

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In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class 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.

Definition of a monotone class

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A monotone class is a family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:

  1. if and then and
  2. if and then

Monotone class theorem for sets

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Monotone class theorem for sets — Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the 𝜎-algebra generated by ; that is

Monotone class theorem for functions

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Monotone class theorem for functions — Let be a π-system that contains and let be a collection of functions from to with the following properties:

  1. If then where denotes the indicator function of
  2. If and then and
  3. 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 𝜎-algebra generated by

Proof

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The following argument originates in Rick Durrett's Probability: Theory and Examples.[1]

Proof

The assumption (2), and (3) imply that is a 𝜆-system. By (1) and the π−𝜆 theorem, Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to

Results and applications

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As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-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

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  • Dynkin system – Family closed under complements and countable disjoint unions
  • π-𝜆 theorem – Family closed under complements and countable disjoint unions
  • π-system – Family of sets closed under intersection
  • σ-algebra – Algebraic structure of set algebra

Citations

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  1. ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.

References

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