My aim is to understand the usual augmentation filtration, for example as given in the appendix in the book Financial Derivatives in Theory and Practice: Revised Edition by P.J. Hunt and J.E. Kennedy.
Let $(\Omega,{\cal F}^\circ,\mathbb{P})$ be a probability triple. The $\mathbb{P}$-completion $(\Omega,{\cal F},\mathbb{P})$ of $(\Omega,{\cal F}^\circ,\mathbb{P})$ is the probability triple defined by ${\cal F}=\sigma({\cal F}^\circ \cup {\cal N})$, where $${\cal N}:=\{A\subset \Omega: A\subseteq B\mbox{ for some }B\in{\cal F}^\circ\mbox{ with }\mathbb{P}(B)=0\}.$$
The usual augmentation $(\Omega,\{{\cal F}_t\},{\cal F},\mathbb{P})$ of the filtered probability space $(\Omega,\{{\cal F}^\circ_t\},{\cal F}^\circ,\mathbb{P})$ is produced by taking ${\cal F}$ to be the $\mathbb{P}$-completion of ${\cal F}^\circ$ and by defining $${\cal F}_t:=\sigma({\cal F}^\circ_{t+}\cup {\cal N}) =\bigcap_{s>t} \sigma({\cal F}^\circ_{s}\cup {\cal N})$$ for all $t\geq 0$.
From this, I have a question:
Do we have the following identity?
$$\sigma(\cap_{i \ge 1}C_i)=\cap_{i \ge 1}\sigma(C_i)$$
Here $C_i$ is a family of subsets of $\Omega$
and $C_1 \supset C_2 \supset C_3....$ is a subset of a set Ω.