PlanetMath: monotone class theorem

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monotone class theorem

(Theorem)

Theorem 1 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 $\mathcal{M}_A=\{B \in \mathcal{M}\vert A \cap B, A \cap B^\complement$ 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 $A \cap B, A \cap B^\complement$ 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 $A \cap B, A \cap B^\complement$ and $A^\complement \cap B \in \mathcal{M}$ , and, of course, $\Omega \in \mathcal{M}$ . $\qedsymbol$

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

$\displaystyle P(X_1 \in A_1, X_2 \in A_2,...,X_n \in A_n)=P(X_1 \in A_1)P(X_2 \in A_2)\cdots P(X_n \in A_n)$

for any Borel sets

and any finite

. 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

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

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

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

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 $B \in \sigma(X_{n+1},X_{n+2},...)$

"monotone class theorem" is owned by fernsanz.

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See Also: monotone class, $\sigma$ -algebra, algebra

Keywords:

algebra, sigma algebra, monotone class

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Cross-references: relation, rectangles, measurable, event, finite, Borel sets, random variables, independent, sequence, infinite, generating, property, applications, field, clear, generated by, sigma algebra, monotone class, subsets, algebra
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This is version 5 of monotone class theorem, born on 2007-05-21, modified 2007-12-16.
Object id is 9429, canonical name is MonotoneClassTheorem.
Accessed 1391 times total.

Classification:

AMS MSC:

28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)

Pending Errata and Addenda

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