Atom in a countably-generated \sigma-algebra

Question

The atoms in a countably generated in a $\sigma$-algebra $\mathcal{A} $ is given by $$\big[ x \big]_{\mathcal{A}} =\underset{x \in A, A \in \mathcal{A} }{\bigcap} A= \underset{x \in A_{i} }{\bigcap} A_{i} \cap \underset{ x \notin A_{i}} {\bigcap X } \setminus A_{i}.$$ One direction is clear but I am really confused for the opposite one. I would be grateful if someone could help me.

Answer 1

Consider the elements $B_I:=\bigcap_{i\in I}A_i\, \cap\, \bigcap_{i\notin I} X\setminus A_i$ for any $I\subseteq \Bbb N$. Observe that these are pairwise disjoint, and that $\bigcup_I B_I=X$.

Then define $$\mathcal B:=\left\{\bigcup_{I\in \mathcal J} B_I\, :\, \mathcal J\subseteq P(\Bbb N) \, \text{ countable or co-countable}\right\}$$ and prove that it's a $\sigma$-algebra, hence it is the $\sigma$-algebra generated by the sets $A_i$, and as every element of it is a union of $B_I$'s, the nonempty ones must be atoms.