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 $G$ 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 family (i.e. class) $M$ of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means $M$ has the following properties:

1. if $A_1,A_2,\dots \in M$ and $A_1\subseteq A_2\subseteq \cdots$ then $\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in M,$ and
2. if $B_1,B_2,\dots \in M$ and $B_1\supseteq B_2\supseteq \cdots$ then $\bigcap _{i=1}^{\mathrm{\infty }}{B}_i\in M.$

Monotone class theorem for sets

Monotone class theorem for sets—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 $G$; that is $\sigma (G)=M(G).$

Monotone class theorem for functions

Monotone class theorem for functions—Let $𝒜$ be a π-system that contains $\Omega$ and let $\mathscr{H}$ be a collection of functions from $\Omega$ to $ℝ$ with the following properties:

1. If $A\in 𝒜$ then ${\mathrm{𝟏}}_A\in \mathscr{H}$ where ${\mathrm{𝟏}}_A$ denotes the indicator function of $A.$
2. If $f,g\in \mathscr{H}$ and $c\in ℝ$ then $f+g$ and $cf\in \mathscr{H}.$
3. If $f_n\in \mathscr{H}$ is a sequence of non-negative functions that increase to a bounded function $f$ then $f\in \mathscr{H}.$

Then $\mathscr{H}$ contains all bounded functions that are measurable with respect to $\sigma (𝒜),$ which is the 𝜎-algebra generated by $𝒜.$

Proof

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

Proof

The assumption $\Omega \in 𝒜,$ (2), and (3) imply that $𝒢=\{A:{\mathrm{𝟏}}_A\in \mathscr{H}\}$ is a 𝜆-system. By (1) and the π−𝜆 theorem, $\sigma (𝒜)\subseteq 𝒢.$ Statement (2) implies that $\mathscr{H}$ contains all simple functions, and then (3) implies that $\mathscr{H}$ contains all bounded functions measurable with respect to $\sigma (𝒜).$

Results and applications

As a corollary, if $G$ is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of $G.$

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-𝜆 theorem – Family closed under complements and countable disjoint unionsPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• σ-algebra – Algebraic structure of set algebra