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 in a set $R$ is a collection $M$ of subsets of $R$ which contains $R$ and is closed under countable monotone unions and intersections, i.e. if $A_i \in M$ and $A_1 \subset A_2 \subset \ldots$ then $\cup_{i = 1}^\infty A_i \in M$ , and similarly for intersections of decreasing sequences of sets.

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. ^[1]

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 π − λ 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.

References

↑ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-0521765398.

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