In Real Analysis by Folland,
the product σ-algebra $ \displaystyle \bigotimes_{\alpha \in A} \mathcal{M_{\alpha}} $ on $X$ is defined by the σ-algebra generated by $ \{ \pi ^{-1} _{\alpha} (E_{\alpha}): E_{\alpha} \in \mathcal{M_{\alpha}} \} $
where $ \pi _{\alpha} : \displaystyle \prod_{\alpha \in A} X_{\alpha} =: X \rightarrow X_{\alpha}$ is a coordinate map.
Prop 1.3: If $A$ is countable, then $ \displaystyle \bigotimes_{\alpha \in A} \mathcal{M_{\alpha}} $ is the σ-algebra generated by $ \{ \displaystyle \prod_{\alpha \in A} E_{\alpha} : E_{\alpha} \in \mathcal{M_\alpha} \} $
I want to find a counter example that if A is uncountable, then the above proposition doesn't hold.. but I'm totally stuck, please help! Thanks in advance.