The description of the product sigma-algebra for countable products This is taken from Folland 's analysis book. 


I basically got stuck on understanding the proof of proposition 1.3 .
I will appreciate if anyone can help to clarify.
P/S : Can anybody show me how to type math notation on this site ?
 A: Late answer, but there is an error in your print of the book. In the first line of the proof of proposition 1.3, $E_{\beta} = X$ should be $E_{\beta} = X_{\beta}$.
A: The proof, with more details. Let $\mathcal{E}$ be the generating set that defines the product sigma algebra. Let $\mathcal{F}$ be the set in the proposition. 
Firstly, notice that if we take an element of $\mathcal{E}$, it's an element of $\mathcal{F}$, hence an element of $\mathcal{M}(\mathcal{F})$. So Lemma 1.1 tells us that $\mathcal{M}(\mathcal{E})$ is contained in $\mathcal{M}(\mathcal{F})$.
Now, notice that if we take an element of $\mathcal{F}$, it's a countable intersection of elements of $\mathcal{E}$, hence an element of $\mathcal{M}(\mathcal{E})$. The equality follows by the definition of the product in terms of projections - i.e. that an element $x$ is in the product on the LH side if and only if for every $\alpha$, $\pi_\alpha(x)$ is in $E_\alpha$. So Lemma 1.1 tells us that $\mathcal{M}(\mathcal{F})$ is contained in $\mathcal{M}(\mathcal{E})$.
Therefore, $\mathcal{M}(\mathcal{F})$ = $\mathcal{M}(\mathcal{E})$, which is exactly what we wanted.
