# 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 similarly in the other direction.

## Monotone class theorem for sets

### Statement

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

## Monotone class theorem for functions

### Statement

Let be a π-system that contains and let be a collection of functions from to **R** with the following properties:

(1) If , then

(2) If , then and for any real number

(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 , the sigma-algebra generated by

### Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]

The assumption , (2) and (3) imply that is a *λ*-system. By (1) and the π−*λ* theorem, . (2) implies contains all simple functions, and then (3) 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.

## References

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