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

(1) If $A\in {\mathcal {A}}$ , then $\mathbf {1} _{A}\in {\mathcal {H}}$ (2) If $f,g\in {\mathcal {H}}$ , then $f+g$ and $cf\in {\mathcal {H}}$ for any real number $c$ (3) If $f_{n}\in {\mathcal {H}}$ is a sequence of non-negative functions that increase to a bounded function $f$ , then $f\in {\mathcal {H}}$ Then ${\mathcal {H}}$ contains all bounded functions that are measurable with respect to $\sigma ({\mathcal {A}})$ , the sigma-algebra generated by ${\mathcal {A}}$ Proof

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

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

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.