Monotone class theorem

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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