Proof of the Monotone Class Theorem

Question

I am learning the Monotone Class Theorem from Jacod's Probability Essentials. I don't quite understand the idea of the proof in the book.

I don't see the point in the proof at all. What's the use of the "magic" construction $\mathcal{B}_B$? It's only clear to me that

• $\Omega\in\mathcal{B}$;
• $\mathcal{B}$ is closed under complement since it's closed by set difference and $\Omega\in\mathcal{B}$;

What's left is to show

1. $\mathcal{B}$ is closed under countable union;
2. $\mathcal{B}$ is the smallest $\sigma$-algebra containing $\mathcal{C}$.

However, what's the logical order of the proof? In which part of the proof are these two points shown?

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[EDIT:]I think 1 is a direct result of the assumption that $\mathcal{B}$ is closed under increasing limits. For any $\{B_n\}_{n=1}^{\infty}\subset\mathcal{B}$, $\{\cup_{i=1}^{n}B_i\}_{n=1}^{\infty}\subset\mathcal{B}$ is increasing and $$ \cup_{n=1}^{\infty}B_n=\cup_{n=1}^{\infty}{\tilde B}_n, {\tilde B}_n:=\cup_{i=1}^{n}B_i. $$

Answer 1

Since any $\sigma$-algebra containing $\mathcal{C}$ is closed under increasing limit and set difference, such $\sigma$-algebra must contain $\mathcal{B}$ by definition of $\mathcal{B}$. If $\mathcal{B}$ is a $\sigma$-algebra containing $\mathcal{C}$, then it must be the smallest one. Therefore the main work for the proof is showing that $\mathcal{B}$ is a $\sigma$-algebra, which means

1. $\Omega\in\mathcal{B}$;
2. $\mathcal{B}$ is closed under complement;
3. $\mathcal{B}$ is closed under countable union.

According to OP, it remains to show 3. For any $\{B_n\}_{n=1}^{\infty}\subset\mathcal{B}$, $\{\cup_{i=1}^{n}B_i\}_{n=1}^{\infty}\subset\mathcal{B}$ is increasing and
$$ \bigcup_{n=1}^{\infty}B_n=\bigcup_{n=1}^{\infty}{\tilde B}_n, \quad {\tilde B}_n:=\bigcup_{i=1}^{n}B_i. $$ It suffices to show that ${\tilde B}_n:=\cup_{i=1}^{n}B_i\in\mathcal{B}$. It is for this reason that one needs $\mathcal{B}_B$.