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 thesmallest 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 algebrawhich 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, infact, ${ℳ}_A=ℳ$, 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 thatfor 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 ${ℱ}_0\subset {ℳ}_B$ andagain 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 thatof showing that certain property is satisfied by all sets in an$\sigma$-algebra, generally starting by the fact that the fieldgenerating the $\sigma$-algebra satisfies such property and that thesets 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 anyevent 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 andB are measurable rectangles in ${ℬ}^n$ and ${ℬ}^{\mathrm{\infty }}$ respectively. Now it is clear that the sets A whichsatisfies 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 measurablerectangle $B\in {ℬ}^{\mathrm{\infty }}$. A second application of thetheorem shows finally that the above relation holds for any $A\in \sigma (X_1,X_2,\mathrm{\dots },X_n)$ and $B\in \sigma (X_{n+1},X_{n+2},\mathrm{\dots })$