# monotone class theorem

Monotone Class theoremFernando Sanz Gamiz

###### Theorem.

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

###### Proof.

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

Let $\mathcal{M}_{A}=\{B\in\mathcal{M}|A\cap B,A\cap B^{\complement}\mbox{ and }A^{% \complement}\cap B\in\mathcal{M}\}$. Then is clear that $\mathcal{M}_{A}$ is a monotone class and, in fact, $\mathcal{M}_{A}=\mathcal{M}$, for if $A\in\mathcal{F}_{0}$, then $\mathcal{F}_{0}\subset\mathcal{M}_{A}$ since $\mathcal{F}_{0}$ is a field, hence $\mathcal{M}\subset\mathcal{M}_{A}$ by minimality of $\mathcal{M}$; consequently $\mathcal{M}=\mathcal{M}_{A}$ by definition of $\mathcal{M}_{A}$. But this shows that for any $B\in\mathcal{M}$ we have $A\cap B,A\cap B^{\complement}\mbox{ and }A^{\complement}\cap B\in\mathcal{M}$ for any $A\in\mathcal{F}_{0}$, so that $\mathcal{F}_{0}\subset\mathcal{M}_{B}$ and again by minimality $\mathcal{M}=\mathcal{M}_{B}$. But what we have just proved is that $\mathcal{M}$ is an algebra, for if $A,B\in\mathcal{M}=\mathcal{M}_{A}$ we have showed that $A\cap B,A\cap B^{\complement}\mbox{ and }A^{\complement}\cap B\in\mathcal{M}$, and, of course, $\Omega\in\mathcal{M}$. ∎

###### 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

 $P(X_{1}\in A_{1},X_{2}\in A_{2},...,X_{n}\in A_{n})=P(X_{1}\in A_{1})P(X_{2}% \in A_{2})\cdots P(X_{n}\in A_{n})$

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 $\sigma(X_{1},X_{2},...,X_{n})$ is independent of any event in $\sigma(X_{n+1},X_{n+2},...)$. For, by independence

 $P((X_{1},X_{2},...,X_{n})\in A,(X_{n+1},X_{n+2},...)\in B)=P((X_{1},X_{2},...,% X_{n})\in A)P((X_{n+1},X_{n+2},...)\in B)$

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

 $P((X_{1},X_{2},...,X_{n})\in A,(X_{n+1},X_{n+2},...)\in B)=P((X_{1},X_{2},...,% X_{n})\in A)P((X_{n+1},X_{n+2},...)\in B)$

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

Title monotone class theorem MonotoneClassTheorem 2013-03-22 17:07:34 2013-03-22 17:07:34 fernsanz (8869) fernsanz (8869) 8 fernsanz (8869) Theorem msc 28A05 MonotoneClass SigmaAlgebra Algebra FunctionalMonotoneClassTheorem