Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets [math]\displaystyle{ G }[/math] is precisely the smallest 𝜎-algebra containing [math]\displaystyle{ G. }[/math] 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 family (i.e. class) [math]\displaystyle{ M }[/math] of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means [math]\displaystyle{ M }[/math] has the following properties:

1. if [math]\displaystyle{ A_1, A_2, \ldots \in M }[/math] and [math]\displaystyle{ A_1 \subseteq A_2 \subseteq \cdots }[/math] then [math]\displaystyle{ {\textstyle\bigcup\limits_{i = 1}^\infty} A_i \in M, }[/math] and
2. if [math]\displaystyle{ B_1, B_2, \ldots \in M }[/math] and [math]\displaystyle{ B_1 \supseteq B_2 \supseteq \cdots }[/math] then [math]\displaystyle{ {\textstyle\bigcap\limits_{i = 1}^\infty} B_i \in M. }[/math]

Monotone class theorem for sets

Monotone class theorem for sets — Let [math]\displaystyle{ G }[/math] be an algebra of sets and define [math]\displaystyle{ M(G) }[/math] to be the smallest monotone class containing [math]\displaystyle{ G. }[/math] Then [math]\displaystyle{ M(G) }[/math] is precisely the 𝜎-algebra generated by [math]\displaystyle{ G }[/math]; that is [math]\displaystyle{ \sigma(G) = M(G). }[/math]

Monotone class theorem for functions

Monotone class theorem for functions — Let [math]\displaystyle{ \mathcal{A} }[/math] be a π-system that contains [math]\displaystyle{ \Omega\, }[/math] and let [math]\displaystyle{ \mathcal{H} }[/math] be a collection of functions from [math]\displaystyle{ \Omega }[/math] to [math]\displaystyle{ \R }[/math] with the following properties:

1. If [math]\displaystyle{ A \in \mathcal{A} }[/math] then [math]\displaystyle{ \mathbf{1}_A \in \mathcal{H} }[/math] where [math]\displaystyle{ \mathbf{1}_A }[/math] denotes the indicator function of [math]\displaystyle{ A. }[/math]
2. If [math]\displaystyle{ f, g \in \mathcal{H} }[/math] and [math]\displaystyle{ c \in \Reals }[/math] then [math]\displaystyle{ f + g }[/math] and [math]\displaystyle{ c f \in \mathcal{H}. }[/math]
3. If [math]\displaystyle{ f_n \in \mathcal{H} }[/math] is a sequence of non-negative functions that increase to a bounded function [math]\displaystyle{ f }[/math] then [math]\displaystyle{ f \in \mathcal{H}. }[/math]

Then [math]\displaystyle{ \mathcal{H} }[/math] contains all bounded functions that are measurable with respect to [math]\displaystyle{ \sigma(\mathcal{A}), }[/math] which is the 𝜎-algebra generated by [math]\displaystyle{ \mathcal{A}. }[/math]

Proof

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

Proof

The assumption [math]\displaystyle{ \Omega\, \in \mathcal{A}, }[/math] (2), and (3) imply that [math]\displaystyle{ \mathcal{G} = \left\{A : \mathbf{1}_{A} \in \mathcal{H}\right\} }[/math] is a 𝜆-system. By (1) and the π−𝜆 theorem, [math]\displaystyle{ \sigma(\mathcal{A}) \subseteq \mathcal{G}. }[/math] Statement (2) implies that [math]\displaystyle{ \mathcal{H} }[/math] contains all simple functions, and then (3) implies that [math]\displaystyle{ \mathcal{H} }[/math] contains all bounded functions measurable with respect to [math]\displaystyle{ \sigma(\mathcal{A}). }[/math]

Results and applications

As a corollary, if [math]\displaystyle{ G }[/math] is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of [math]\displaystyle{ G. }[/math]

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-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.

See also

• Dynkin system – Family closed under complements and countable disjoint unions
• π-system – Family of sets closed under intersection
• Σ-algebra – Algebraic structure of set algebra

References

• Template:Durrett Probability Theory and Examples 5th Edition

fr:Lemme de classe monotone