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

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

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

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

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

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

  • 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

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

    References

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