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I found different statements of the Monotone Class Theorem.

On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows:

Let $\mathcal{C}$ be a class of subsets of $\Omega$ under finite intersections and containing $\Omega$. Let $\mathcal{B}$ be the smallest class containing $\mathcal{C}$ which is closed under increasing limits and by difference. Then $\mathcal{B} = \sigma ( \mathcal{C})$.

While on Wikipedia (https://en.wikipedia.org/wiki/Monotone_class_theorem) the theorem is:

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Let $G$ be an algebra of sets and define $M(G)$ to be the smallest monotone class containing $G$. Then $M(G)$ is precisely the $\sigma$-algebra generated by $G$, i.e. $\sigma(G) = M(G)$.

Where a monotone class in a set $R$ is a collection $M$ of subsets of $R$ which contains $R$ and is closed under countable monotone unions and intersections.

It looks like the second theorem should be a special case of the first. Does the first prove the second? Is it possible to prove the first from the second? Is there a decent literature on those two theorems?

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Both results are actually equivalent. You can prove one from the other.

Regarding the first result:

Let $\mathcal{C}$ be a class of subsets of $\Omega$ under finite intersections and containing $\Omega$. Let $\mathcal{B}$ be the smallest class containing $\mathcal{C}$ which is closed under increasing limits and by difference. Then $\mathcal{B} = \sigma ( \mathcal{C})$.

Some books call it “Monotone Class Theorem”, although this is not the most usual naming.

A class having $\Omega$, closed under increasing limits and by difference is called a “Dynkin $\lambda$ system”. A non-empty class closed under finite intersections is called a “Dynkin $\pi$ system”.

The result above can be divided in two results

1.a. A $\lambda$ system which is also a $\pi$ system is a $\sigma$-algebra.

1.b. Given a $\pi$ system, the smallest $\lambda$ system containing it is also a $\pi$ system.

Some books call result 1 (or result 1.b.) “Dynkin $\pi$-\lambda$ Theorem.

Some quick references is

https://en.wikipedia.org/wiki/Dynkin_system

The second result

Let $G$ be an algebra of sets and define $M(G)$ to be the smallest monotone class containing $G$. Then $M(G)$ is precisely the $\sigma$-algebra generated by $G$, i.e. $\sigma(G) = M(G)$.

Where a monotone class in a set $R$ is a collection $M$ of subsets of $R$ which contains $R$ and is closed under countable monotone unions and intersections.

is usually called “Monotone Class Lemma” (or theorem) you can find it in books like Folland’s *Real Analysis* or Halmos’ *Measure Theory*. In fact, Halmos presents a version of this result for $\sigma$-rings.

Let $G$ be ring of sets and define $M(G)$ to be the smallest monotone class containing $G$. Then $M(G)$ is precisely the $\sigma$-ring generated by $G$.

**Let us prove that the results are equivalent**

Result 1: Let $\mathcal{C}$ be a class of subsets of $\Omega$ under finite intersections and containing $\Omega$. Let $L(\mathcal{C})$ be the smallest class containing $\mathcal{C}$ which is closed under increasing limits and by difference. Then $L(\mathcal{C}) = \sigma ( \mathcal{C})$.

Result 2: Let $G$ be an algebra of sets and define $M(G)$ to be the smallest monotone class containing $G$. Then $M(G)$ is precisely the $\sigma$-algebra generated by $G$, i.e. $\sigma(G) = M(G)$.Where a monotone class in a set $R$ is a collection $M$ of subsets of $R$ which contains $R$ and is closed under countable monotone unions and intersections.

**Proof**:

**(2 $\Rightarrow$ 1)**. Note that any class containing $\mathcal{C}$ which is closed under increasing limits and by difference is close by complement because $\Omega \in \mathcal{C}$, and so it is also closed by decreasing limits. So it is closed under countable monotone unions and intersections. It means: any class containing $\mathcal{C}$ which is closed under increasing limits and by difference is monotone class.

Note also that any class containing $\mathcal{C}$ which is closed under increasing limits and by difference contains $A(\mathcal{C})$ the algebra generated by $\mathcal{C}$.

Then using Result 2 we have

$$ \sigma(\mathcal{C}) = \sigma(A(\mathcal{C})) = M(A(\mathcal{C})) \subseteq L(A(\mathcal{C}))=L(\mathcal{C}) $$

Since $\sigma(\mathcal{C})$ is a class containing $\mathcal{C}$ which is closed under increasing limits and by difference, we have $L(\mathcal{C}) \subseteq \sigma(\mathcal{C})$, so $L(\mathcal{C}) = \sigma(\mathcal{C})$.

**(1 $\Rightarrow$ 2)**. First let us prove that $M(G)$ is a class containing $G$ which is closed under increasing limits and by difference. Since $M(G)$ is monotone, we have that $M(G)$ is closed under increasing limits.

Now, for each $E\in M(G)$, define

$$M_E=\{ F \in M(G) : E\setminus F , F \setminus E \in M(G) \}$$

Since $M(G)$ is a monotone class, $M_E$ is a monotone class. Moreover, if $E\in G$ then for all $F \in G$, $F\in M_E$, because $G$ is an algebra. So, if $E\in G$, $G \subset M_E$. So, if $E\in G$, $M(G) \subset M_E$. It means that for all $E\in G$, and all $F \in M(G)$, $F \in M_E$. So, for all $E\in G$, and all $F \in M(G)$, $E \in M_F$. So, for all $F \in M(G)$, $G \subset M_F$, but since

$M_F$ is a monotone class, we have, for all $F \in M(G)$, $M(G)\subset M_F$. But that means that $M(G)$ is closed by differences.

So we proved that $M(G)$ is a class containing $G$ which is closed under increasing limits and by difference.

So by Result 1, $$\sigma(G)=L(G) \subseteq M(G)$$

Since $\sigma(G)$ is a monotone class, we have

$$ M(G) \subseteq \sigma(G)$$

So we have $$\sigma(G)= M(G)$$

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