3 the Monotone Class Theorem

Definition

A monotone class is a collection of subsets $\mathcal{M}$ of $X$ such that

• if $A_i\uparrow A$ and each $A_i\in \mathcal{M}$, then $A\in \mathcal{M}$;
• if $A_i\downarrow A$ and each $A_i\in \mathcal{M}$, then $A\in \mathcal{M}$.

Remark. If $\{{\mathcal{J}}_{\alpha }{\}}_{\alpha \in \Lambda }$ is a family of monotone classes, then ${\cap }_{\alpha \in \Lambda }{F}_{\alpha }$ is also a monotone class. If $\mathcal{C}\subseteq \mathcal{P}(X)$, $\mathcal{M}(\mathcal{C})$ is the smallest monotone class containing $\mathcal{C}$, called the monotone class generated by $\mathcal{C}$.

MCT

If ${\mathcal{A}}_0$ is an algebra on $X$, then $\mathcal{M}({\mathcal{A}}_0)=\sigma ({\mathcal{A}}_0)$.

\begin{proof} Note that a $\sigma$-algebra is a monotone class, so $\mathcal{M}({\mathcal{A}}_0)\subseteq \sigma ({\mathcal{A}}_0)$. It suffices to show that $\mathcal{M}({\mathcal{A}}_0)$ is a $\sigma$-algebra.

Let ${\mathcal{M}}_1=\{A\in \mathcal{M}:A^c\in \mathcal{M}\}\subseteq \mathcal{M}$. For any $A_i\downarrow A$ and each $A_i\in {\mathcal{M}}_1$, then $A_i^c\in \mathcal{M}$. Then $A\in \mathcal{M}$ and $A_i^c\uparrow A^c$ yield that $A^c\in \mathcal{M}$. Thus $A^c\in {\mathcal{M}}_1$. Similarly, for any $A_i\uparrow A$ and each $A_i\in {\mathcal{M}}_1$, there is $A\in {\mathcal{M}}_1$. Therefore, ${\mathcal{M}}_1$ is a monotone class. Since ${\mathcal{A}}_0\subseteq {\mathcal{M}}_1$, then ${\mathcal{M}}_1=\mathcal{M}$, that is, for any $A\in \mathcal{M}$, $A^c\in \mathcal{M}$.

Aim:

• if $A\in \mathcal{M}$ and $B\in {\mathcal{A}}_0$, then $A\cap B\in \mathcal{M}$.
• if $A,B\in \mathcal{M}$, then $A\cap B\in \mathcal{M}$.
• if $\{A_i\}\subseteq \mathcal{M}$, then $\cap A_i\in \mathcal{M}$.

i) Let $N_1=\{A\in \mathcal{M}:A\cap B\in \mathcal{M},\forall B\in {\mathcal{A}}_0\}$. Then ${\mathcal{A}}_0\subseteq N_1$. Claim that $N_1$ is a monotone class. Let $\{A_i\}\subseteq N$ with $A_i\downarrow A$. Since $A_i\cap B\in \mathcal{M}$, then $A_i\cap B\downarrow A\cap B$ and $A\cap B\in M$ for all $B\in {\mathcal{A}}_0$. Thus $A\in N_1$. Similarly, if $\{A_i\}\subseteq N_1$ with $A_i\uparrow A$, then $A\in N_1$. So $N_1=\mathcal{M}$.

ii) $N_2=\{A\in \mathcal{M}:A\cap B\in \mathcal{M},\forall B\in \mathcal{M}\}$. Similarly we can show that $N_2$ is a monotone class and so $N_2=\mathcal{M}$.

iii) Let $\{A_i\}\subseteq \mathcal{M}$ and let $B_n={\cap }_{i=1}^n{A}_i$. Then $B_n\downarrow A$ and so $A\in \mathcal{M}$. \end{proof}

Definition

$\mathcal{D}\subseteq \mathcal{P}(X)$ is a $d$-system if

• $X\in \mathcal{D}$;
• if $A,B\in \mathcal{D}$ and $A\subseteq B$, then $B-A\in \mathcal{D}$;
• if $\{A_i\}\subseteq \mathcal{D}$ and $A_i\uparrow A$, then $A\in \mathcal{D}$.

Example. Suppose $\mathcal{A}$ is a $\sigma$-algebra on $X$. Then $\mathcal{A}$ is certainly a $d$-system. Furthermore, if $\mu$ and $\nu$ are finite measures on $\mathcal{A}$ such that $\mu (X)=\nu (X)$, then the collection $\mathcal{S}$ of all sets $A$ that belong to $\mathcal{A}$ and satisfy $\mu (A)=\nu (A)$ is a $d$ system.

Definition

$\mathcal{F}\subseteq \mathcal{P}(X)$ is a $\pi$-system if $A,B\in \mathcal{F}$, then $A\cap B\in \mathcal{F}$.

Remark.

• $d$-system is a monotone class.
• If $\mathcal{C}\subseteq \mathcal{P}(X)$, then $\mathcal{D}(\mathcal{C})$ is the $d$-system generated by $\mathcal{C}$, which is the smallest $d$-system containing $\mathcal{C}$.
• A $\sigma$-algebra is a $d$-system.

Theorem

Let $\mathcal{C}\subseteq \mathcal{P}(X)$ be a $\pi$-system. Then $\mathcal{D}(\mathcal{C})=\sigma (\mathcal{C})$.

\begin{proof} Since $\sigma (\mathcal{C})$ is a $d$-system, then $\sigma (\mathcal{C})\supseteq \mathcal{D}(\mathcal{C})$. It suffices to show $\mathcal{D}(\mathcal{C})$ is a $\sigma$-algebra. By ^b47d33, only need to show $\mathcal{D}(\mathcal{C})$ is an algebra, and only need to show for any $A,B\in \mathcal{D}(C)$, there is $A\cap B\in \mathcal{D}(C)$. $(\ast )$

Let ${\mathcal{D}}_1=\{A\in \mathcal{D}:A\cap B\in \mathcal{D},\forall B\in \mathcal{C}\}$. Since $\mathcal{C}$ is a $\pi$-system, then $\mathcal{C}\subseteq {\mathcal{D}}_1$. We claim that ${\mathcal{D}}_1$ is a $d$-system and so ${\mathcal{D}}_1=\mathcal{D}$.

• $X\in {\mathcal{D}}_1$
• If $A_1,A_2\in {\mathcal{D}}_1$, then $(A_1-A_2)\cap B=(A_1\cap B)-(A_2\cap B)\in \mathcal{D}$ and $A_1-A_2\in {\mathcal{D}}_1$.
• If $\{A_i\}\subseteq {\mathcal{D}}_1$ and $A_1\uparrow A$, then for any $B\in \mathcal{C}$, $A_i\cap B\uparrow A\cap B$ and $A\cap B\in \mathcal{D}$. Thus $A\in {\mathcal{D}}_1$.

Let ${\mathcal{D}}_2=\{A\in \mathcal{D}:A\cap B\in \mathcal{D},\forall B\in \mathcal{D}\}$. Similarly, we can prove that ${\mathcal{D}}_2=\mathcal{D}$.

Now we prove $(\ast )$ and so the proof is completed. \end{proof}

Corollary

Let $(X,\mathcal{A})$ be a measurable space, and let $\mathcal{C}$ be a $\pi$-system on $X$ such that $\mathcal{A}=\sigma (\mathcal{C})$. If $\mu$ and $\nu$ are finite measures on $(X,\mathcal{A})$ that satisfy $\mu (X)=\nu (X)$ and satisfy $\mu (A)=\nu (A)$ for each $A\in \mathcal{C}$. Then $\mu =\nu$.

\begin{proof} By this example, $\mathcal{L}:=\{A\in \mathcal{A}:\mu (A)=\nu (A)\}$ is a $d$-system. Since $\mathcal{C}\subseteq \mathcal{L}$, then $\mathcal{L}\supseteq \mathcal{D}(\mathcal{C})=\sigma (\mathcal{C})=\mathcal{A}$. Thus $\mathcal{L}=\mathcal{A}$ and so $\mu =\nu$. \end{proof}

Corollary

Let $(X,\mathcal{A})$ be a measurable space, and let $\mathcal{C}$ be a $\pi$-system on $X$ such that $\mathcal{A}=\sigma (\mathcal{C})$. If $\mu$ and $\nu$ are measures on $(X,\mathcal{A})$ that agree on $\mathcal{C}$, and if there is an increasing sequence $\left\{C_n\right\}$ of sets that belong to $\mathcal{C}$, have finite measure under $\mu$ and $\nu$, and satisfy ${\cup }_{n=1}^{\infty }{C}_n=X$, then $\mu =\nu$.

\begin{proof} Define ${\mu }_i:\mathcal{A}\to [0,\infty ),{\mu }_i(A)=\mu (A\cap C_i)$, and define ${\nu }_i:\mathcal{A}\to [0,\infty ),\nu (A_i)=\nu (A\cap C_i)$. Then by ^9bd144, we have that ${\mu }_i={\nu }_i$ on $\mathcal{C}$ and ${\mu }_i={\nu }_i$ on $\mathcal{A}$. For any $A\in \mathcal{A}$, there is $A\cap C_i\uparrow A\cap X=A$ and so

$$\mu (A)=\underset{i\to \infty }{\lim }{\mu }_i(A)=\underset{i\to \infty }{\lim }{\nu }_i(A)=\nu (A).$$

\end{proof}

Remark

It is a note for midterm

The proof of MCT and $\pi$-$\lambda$ theorem are similar, with two key points:

• all sets with property A is a B-set, which should be the whole family as the family is the minimal B-set;
• divide this procedure into two parts if necessary.

Monotone Class

A monotone class describe a simpler method to generate a $\sigma$-algebra from an algebra, by ^b47d33.

The proof is similar as $\pi$-$\lambda$ theorem, considering the following $2$ sets:

• $N_1=\{A\in \mathcal{M}:A\cap B\in \mathcal{M},\forall B\in {\mathcal{A}}_0\}$;
• $N_2=\{A\in \mathcal{M}:A\cap B\in \mathcal{M},\forall B\in \mathcal{M}\}$.

Then show both of them are monotone class containing ${\mathcal{A}}_0$.

$d$-system and $\pi$-system

The $d$-system is defined here. Note that $d$-system is weaker than $\sigma$-algebra, because it has property:

• $X,\varnothing \in \mathcal{D}$;
• closed under complements;
• closed under countable unions of pairwise disjoint sets.

Note that a family which is both $d$-system and $\pi$-system is a $\sigma$-algebra.

The major application of $d$-system is $\pi$-$\lambda$ theorem:

Let $\mathcal{C}\subseteq \mathcal{P}(X)$ be a $\pi$-system. Then $\mathcal{D}(\mathcal{C})=\sigma (\mathcal{C})$.

Link to original

Thus a $d$-system contains a $\pi$-system if and only if it contains the 𝜎-algebra generated by that $\pi$-system.

The proof of this theorem is interesting: to show for any $A,B\in \mathcal{D}(\mathcal{C})$ we have $A\cap B\in \mathcal{D}(\mathcal{C})$, consider the following $2$ sets:

• ${\mathcal{D}}_1=\{A\in \mathcal{D}:A\cap B\in \mathcal{D},\forall B\in \mathcal{C}\}$;
• ${\mathcal{D}}_2=\{A\in \mathcal{D}:A\cap B\in \mathcal{D},\forall B\in \mathcal{D}\}$.

Since ${\mathcal{D}}_1$ is a $d$-system, ${\mathcal{D}}_1=\mathcal{D}$ and so $\mathcal{C}\subseteq {\mathcal{D}}_2$. Then it suffices to show ${\mathcal{D}}_2$ is a $d$-system.

^9bd144 and ^ljjfjj are applications of $\pi$-$\lambda$ theorem. For two measures $\mu ,\lambda$ on $X$, since $\mathcal{L}:=\{A\in \mathcal{A}:\mu (A)=\nu (A)\}$ is a $d$-system, we find that measures can be determined by $\pi$-system of $X$, i.e., $\pi$-system is like “socle” of a measure space.