$\exists$ countably generated $\mathcal F$, s.t. $\sigma(\{ \{\omega \}: \omega\in\Omega \}) \subsetneqq \mathcal F \subsetneqq \mathcal B(\Omega)$?

Solution 1:

It is a theorem of Blackwell

Blackwell, David, On a class of probability spaces, Proc. 3rd Berkeley Sympos. Math. Statist. Probability 2, 1-6 (1956). ZBL0073.12301.

Let $(A,\mathscr A)$ be an analytic measurable space, and let $\mathscr A_0$ be a countably-generated sub-$\sigma$-algebra of $\mathscr A$. Then a subset of $A$ belongs to $\mathscr A_0$ if and only if it belongs to $\mathscr A$ and is the union of a family of atoms of $\mathscr A_0$.

A special case of an analytic measurable space is a Polish space with its Borel sets. If $\{x\} \in \mathcal A_0$ for all $x$, then the condition "is the union of a family of atoms of $\mathscr A_0$" holds for all sets. So in that case $\mathscr A_0 = \mathscr A$.