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 ${\displaystyle A_{i}\in M}$ and ${\displaystyle A_{1}\subset A_{2}\subset \cdots }$ then ${\textstyle \bigcup _{i=1}^{\infty }A_{i}\in M}$, 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 ${\displaystyle {\mathcal {A}}}$ be a π-system that contains ${\displaystyle \Omega \,}$ and let ${\displaystyle {\mathcal {H}}}$ be a collection of functions from ${\displaystyle \Omega }$ to R with the following properties:

(1) If ${\displaystyle A\in {\mathcal {A}}}$, then ${\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}}$

(2) If ${\displaystyle f,g\in {\mathcal {H}}}$, then ${\displaystyle f+g}$ and ${\displaystyle cf\in {\mathcal {H}}}$ for any real number ${\displaystyle c}$

(3) If ${\displaystyle f_{n}\in {\mathcal {H}}}$ is a sequence of non-negative functions that increase to a bounded function ${\displaystyle f}$, then ${\displaystyle f\in {\mathcal {H}}}$

Then ${\displaystyle {\mathcal {H}}}$ contains all bounded functions that are measurable with respect to ${\displaystyle \sigma ({\mathcal {A}})}$, the sigma-algebra generated by ${\displaystyle {\mathcal {A}}}$

Proof

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

The assumption ${\displaystyle \Omega \,\in {\mathcal {A}}}$, (2) and (3) imply that ${\displaystyle {\mathcal {G}}=\{A:\mathbf {1} _{A}\in {\mathcal {H}}\}}$ is a λ-system. By (1) and the πλ theorem, ${\displaystyle \sigma ({\mathcal {A}})\subset {\mathcal {G}}}$. (2) implies ${\displaystyle {\mathcal {H}}}$ contains all simple functions, and then (3) implies that ${\displaystyle {\mathcal {H}}}$ contains all bounded functions measurable with respect to ${\displaystyle \sigma ({\mathcal {A}})}$.

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

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