Monotone class theorem


In measure theory and probability, the monotone class theorem connects monotone classes and -algebras. The theorem says that the smallest monotone class containing an algebra of sets is precisely the smallest -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 is a family 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 functions

Proof

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

Results and applications

As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the -ring of
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a -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.