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
- $\mathcal{B}$ is closed under countable union;
- $\mathcal{B}$ 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?
[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. $