monotone class theorem

Monotone Class theoremFernando Sanz Gamiz

Theorem.

Let ${\mathrm{F}}_{\mathrm{0}}$ an algebra of subsets of $\mathrm{\Omega }$. Let $\mathrm{M}$ be the smallest monotone class such that ${\mathrm{F}}_{\mathrm{0}}\mathrm{\subset }\mathrm{M}$ and $\sigma \mathit{}\mathrm{(}{\mathrm{F}}_{\mathrm{0}}\mathrm{)}$ be the sigma algebra generated by ${\mathrm{F}}_{\mathrm{0}}$. Then $\mathrm{M}\mathrm{=}\sigma \mathit{}\mathrm{(}{\mathrm{F}}_{\mathrm{0}}\mathrm{)}$.

Proof.

It is enough to prove that $ℳ$ is an algebra, because an algebra which is a monotone class is obviously a $\sigma$-algebra.

Let ${ℳ}_A=\{B\in ℳ|A\cap B,A\cap B^{\mathrm{\complement }}\text{ and }{A}^{\mathrm{\complement }}\cap B\in ℳ\}$. Then is clear that ${ℳ}_A$ is a monotone class and, in fact, ${ℳ}_A=ℳ$, for if $A\in {ℱ}_0$, then ${ℱ}_0\subset {ℳ}_A$ since ${ℱ}_0$ is a field, hence $ℳ\subset {ℳ}_A$ by minimality of $ℳ$; consequently $ℳ={ℳ}_A$ by definition of ${ℳ}_A$. But this shows that for any $B\in ℳ$ we have $A\cap B,A\cap B^{\mathrm{\complement }}\text{ and }{A}^{\mathrm{\complement }}\cap B\in ℳ$ for any $A\in {ℱ}_0$, so that ${ℱ}_0\subset {ℳ}_B$ and again by minimality $ℳ={ℳ}_B$. But what we have just proved is that $ℳ$ is an algebra, for if $A,B\in ℳ={ℳ}_A$ we have showed that $A\cap B,A\cap B^{\mathrm{\complement }}\text{ and }{A}^{\mathrm{\complement }}\cap B\in ℳ$, and, of course, $\mathrm{\Omega }\in ℳ$. ∎

Remark 1.

One of the main applications of the Monotone Class Theorem is that of showing that certain property is satisfied by all sets in an $\sigma$-algebra, generally starting by the fact that the field generating the $\sigma$-algebra satisfies such property and that the sets that satisfies it constitutes a monotone class.

Example 1.

Consider an infinite sequence of independent random variables $\{X_n,n\in \mathbb{N}\}$. The definition of independence is

$$P(X_1\in A_1,X_2\in A_2,\mathrm{\dots },X_n\in A_n)=P(X_1\in A_1)P(X_2\in A_2)\mathrm{\cdots }P(X_n\in A_n)$$

for any Borel sets $A_1,A_2,..,A_n$ and any finite $n$. Using the Monotone Class Theorem one can show, for example, that any event in $\sigma (X_1,X_2,\mathrm{\dots },X_n)$ is independent of any event in $\sigma (X_{n+1},X_{n+2},\mathrm{\dots })$. For, by independence

$$P((X_1,X_2,\mathrm{\dots },X_n)\in A,(X_{n+1},X_{n+2},\mathrm{\dots })\in B)=P((X_1,X_2,\mathrm{\dots },X_n)\in A)P((X_{n+1},X_{n+2},\mathrm{\dots })\in B)$$

when A and B are measurable rectangles in ${ℬ}^n$ and ${ℬ}^{\mathrm{\infty }}$ respectively. Now it is clear that the sets A which satisfies the above relation form a monotone class. So

$$P((X_1,X_2,\mathrm{\dots },X_n)\in A,(X_{n+1},X_{n+2},\mathrm{\dots })\in B)=P((X_1,X_2,\mathrm{\dots },X_n)\in A)P((X_{n+1},X_{n+2},\mathrm{\dots })\in B)$$

for every $A\in \sigma (X_1,X_2,\mathrm{\dots },X_n)$ and any measurable rectangle $B\in {ℬ}^{\mathrm{\infty }}$. A second application of the theorem shows finally that the above relation holds for any $A\in \sigma (X_1,X_2,\mathrm{\dots },X_n)$ and $B\in \sigma (X_{n+1},X_{n+2},\mathrm{\dots })$

Title	monotone class theorem
Canonical name	MonotoneClassTheorem
Date of creation	2013-03-22 17:07:34
Last modified on	2013-03-22 17:07:34
Owner	fernsanz (8869)
Last modified by	fernsanz (8869)
Numerical id	8
Author	fernsanz (8869)
Entry type	Theorem
Classification	msc 28A05
Related topic	MonotoneClass
Related topic	SigmaAlgebra
Related topic	Algebra
Related topic	FunctionalMonotoneClassTheorem