I am given an exercise, the following is the first part of the exercise:
let $X_{\alpha}$ be a measureable space with $\sigma-algebra$ $M_{\alpha}$ , mark $X\triangleq{\displaystyle \prod_{\alpha\in A}X_{\alpha}}$ and $\pi_{\alpha}:X\rightarrow X_{\alpha}$ the projection to the i-th coordinate.
define $\otimes_{\alpha\in A}M_{\alpha}$ as the $\sigma-algebra$ that is created from sets of form $\pi_{\alpha}^{-1}\left(E_{\alpha}\right)$ where $E_{\alpha}\in M_{\alpha}$
Prove that if $A$ is countable then $\otimes_{\alpha}M_{\alpha}$ is created by sets of form ${\displaystyle \prod_{\alpha\in A}E_{\alpha}}$
I don't know where to start, I numbered the elements of $A$ and I wanted to prove by showing two containments, but I don't know how to start any of them.
Can someone please give some hint on how to start ?
Edit: I have managed to prove the containment that what created by the sets of form ${\displaystyle \prod_{\alpha\in A}E_{\alpha}}$ are in whats created from sets of form $\pi_{\alpha}^{-1}\left(E_{\alpha}\right)$