Let $\{A_1,A_2,...,A_m\}$ and $\{B_1,B_2,...,B_n\}$ be two partitions of $\Omega$. I am searching a proof of the following facts:
a) $\{A_i \cap B_j\}$ is a partition of $\Omega$.
b) $\sigma\{A_i \cap B_j\}=\sigma\{\{A_i\} \cup \{B_j\}\}=\sigma\{\sigma\{A_i\} \cup \sigma\{B_j\}\}$
I already managed to proof a) and the inclusions $\sigma\{A_i \cap B_j\}\subseteq\sigma\{\{A_i\} \cup \{B_j\}\}$,$\sigma\{\{A_i\} \cup \{B_j\}\}\subseteq\sigma\{\sigma\{A_i\} \cup \sigma\{B_j\}\}$,
But I have problems to prove: $\sigma\{\sigma\{A_i\} \cup \sigma\{B_j\}\}\subseteq\sigma\{A_i \cap B_j\}.$
Note: here, $\sigma \{ A_i \}$ refers to the $\sigma$-algebra generated by $\{ A_i \}$ in $\Omega$; ditto for $\sigma\{ B_j \}$, $\sigma \{ A_i \cap B_j \}$, etc.
Please, someone help me with this.
Hint: Prove that for every $\mathcal{A} \subset \mathcal{P}(\Omega)$, where $\mathcal{P}(\Omega)$ is the power set of $\Omega$, one has $$\sigma\{ \sigma \{ \mathcal{A} \} \} = \sigma\{ \mathcal{A} \}.$$