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 G is precisely the smallest 𝜎-algebra containing G. 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**** (i.e. class)**** M of**** sets**** that**** is**** closed**** under countable monotone**** unions**** and also**** under countable monotone**** intersections.**** Explicitly, this means M has the following properties:

Monotone class theorem for sets

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 G is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of G. 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.

Citations