Monotone class theorem


In measure theory and probability, the monotone class theorem connects monotone classes and sigma-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 monotone class is a class M of sets that is closed under countable monotone unions and intersections, i.e. if and then, and if and then

Monotone class theorem for sets

Statement

Let G be an algebra of sets and define M to be the smallest monotone class containing G. Then M is precisely the σ-algebra generated by G, i.e. σ = M.

Monotone class theorem for functions

Statement

Let be a -system that contains and let be a collection of functions from to R with the following properties:
If, then
If, then and for any real number
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, the sigma-algebra generated by

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.
The assumption, and imply that is a λ-system. By and the −λ theorem,. implies contains all simple functions, and then implies that contains all bounded functions measurable with respect to.

Results and applications

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