Let $\{ X_{\alpha} \}_{\alpha \in A}$ be an indexed collection of nonempty sets, $X = \prod_{\alpha \in A}X_{\alpha}$, and $\pi_{\alpha}: X \rightarrow X_{\alpha}$ the coordinate maps. If $\mathcal{M}_{\alpha}$ is a $\sigma$-algebra on $X_{\alpha}$ for each $\alpha$, the $\textbf{product $\sigma$-algebra}$ on $X$ is the $\sigma$-algebra generated by : $$\{ \pi^{-1}_{\alpha}(E_{\alpha}):E_{\alpha} \in \mathcal{M}_{\alpha}, \alpha \in A \}.$$
Then we have these next two propositions:
1.If $A$ is countable, then $\textbf{product $\sigma$-algebra}$ is generated by $\{\prod_{\alpha \in A} E_{\alpha}:E_{\alpha} \in \mathcal{M}_{\alpha} \}.$
- Suppose $\mathcal{M}_{\alpha}$ is generated by $\mu_{\alpha},\alpha \in A$. Then $\textbf{product $\sigma$-algebra}$ is generated by $$\{ \pi^{-1}_{\alpha}(E_{\alpha}):E_{\alpha} \in \mu_{\alpha}, \alpha \in A \}.$$ Furthermore, if $A$ is countable and $X_{\alpha} \in \mu_{\alpha}, \alpha \in A$, then $\textbf{product $\sigma$-algebra}$ is generated by $\{\prod_{\alpha \in A} E_{\alpha}:E_{\alpha} \in \mu_{\alpha} \}.$
The proof is easy. The first bit hinges on the observation that $$\prod_{\alpha \in A} E_{\alpha}=\cap_{\alpha \in A}\pi^{-1}_{\alpha}(E_{\alpha})$$ and if $E \subset \sigma(F)$(The sigma algebra generated by F), then $\sigma(E) \subset \sigma(F)$.
The first part of second bit follows by constructing a set with the required property and the lemma (if $E \subset \sigma(F)$(The sigma algebra generated by F), then $\sigma(E) \subset \sigma(F))$, The second bit follows from the first bit along with the use of lemma.
Well I was wondering what happens when $A$ is uncountable. Clearly the proof of $1$ won't go through. I was not able to get any counterexample though. I was thinking of taking $A=\mathbb{R}$ and $X_{\alpha}=[0,1]$ for each $\alpha$. But couldn't succeed.
Similarly the proof of $2$ won't go through as well. Can I have a counterexample for this as well?? Also what happens when $X_{\alpha} \not\in \mu_{\alpha}$?? This is certainly used in the proof.
Thanks for the help!!
You're looking for the idea of countable support. Basically, since each basis element $\pi_\alpha^{-1}(E)$ has only one "coordinate" nontrivial and each operation (complement, intersection, union) has at most countable arity, we should expect to only be able to generate sets with countably many "nontrivial coordinates."
Specifically, suppose $Y\subseteq \prod_{\alpha\in A} X_\alpha$ is an element of the product $\sigma$-algebra. Then there is a countable set $B\subset A$ such that for every $y\in Y$ and every $y'$ such that $y'$ and $y$ agree on all coordinates in $B$, $y'$ is also in $Y$. (That is, whether or not something is an element of $Y$ depends on only finitely many coordinates.) Thinking along these lines will help give a better description of the product $\sigma$-algebra, as well as its generating sets.