Does finite intersection and union commute for sigma-fields?

Question

A specific question I want to ask is this:

Let $F, G, H$ be sigma fields, and $K$ be the sigma field generated by $G$ and $H$.

Then, is it true that $F\cap K$ is equal to the sigma field generated by $\left((F \cap G) \cup (F \cap H)\right)$?

Answer 1

1. $G=\{ \emptyset , A, A^c ,\Omega\}$ with $A \subset \Omega$.
2. $H=\{\emptyset, B,B^c,\Omega\}$ with $B\notin \{A,A^c\}$.
3. We have $K=\{\emptyset,A,B,A^c,B^c,A\cap B, A^c\cap B, A\cap B^c, A^c\cap B^c, \dots,\Omega\}.$
4. $F=K$.
5. $ F\cap K = K$.
6. $ (F\cap G)\cup (F\cap H)=\{\emptyset,A,A^c,B,B^c,\Omega\}.$