Suppose that $X, Y$ are given non empty sets. Let $C, D$ be collection of subsets of $X$ and $Y$ respectively. Suppose that there exists sequences $\{C_n\}$ (of elements of $C$) and $\{D_n\}$ (of elements of $D$) such that $\cup_n C_n= X, \cup_n D_n=Y$. Then, $M(C\times D)=M(C)\otimes M(D)$, where $M$('set') means "$\sigma$ algebra generated by the set." $M(C)\otimes M(D)$ is the sigma algebra generated by $M(C)\times M(D)$.
I tried to prove it like this:
It is clear that $M(C\times D)\subset M(C)\otimes M(D)$. For containment in the other direction, define
$\scr E:$$=\{E\in M(C)\otimes M(D): E\in M(C\times D)\}$.
$\scr E$ is non empty as it contains $\emptyset$ and $X\times Y$.
$\scr E$ is an algebra: For any $E_1, E_2\in \scr E$, $E_1\cup E_2\in \scr E$ and $E_1^c\in \scr E$.
$\scr E$ is a monotone class. So if it is shown that $\scr E$ contains all $A\times B, A\in M(C), B\in M(D)$, then we are done. But I'm not sure how to go from here and how to use $\cup_n C_n=C, \cup_n D_n=D$? Any hints on this please?