TITLE: Naive Question About Interpreting a Set Notation
QUESTION [1 upvotes]: I was reading Folland's real analysis and notice that there is one very basic concept about the set notation usage yet I don't understand very well.
In the bottom of page 22, Folland defines a product $\sigma$-algebra on $X$ to be the $\sigma$-algebra generated by the set
  $$
\{ \pi_{\alpha}^{-1}(E_\alpha) : E_\alpha \in \mathcal{M}_\alpha, \alpha \in A \}:=\Pi
$$ 
  where $\pi_{\alpha}: X \to X_\alpha$ is the coordinate maps and $X:=\prod_{\alpha \in A} X_\alpha$ and $\mathcal{M}_{\alpha}$ is the $\sigma$-algebra on $X_\alpha$ for each $\alpha$ and $A$ is the index set.

The thing I got confused is to interpret its (superficial) meaning of the set defined above based on the given form. Say $x \in \Pi$, does this means 
$x$ has the form of $\pi_{\alpha}^{-1}(E_\alpha)$ where $E_{\alpha} \in \mathcal{M}_\alpha$ AND $\alpha \in A$?
If so, may I rewrite directly by saying
$$
\Pi =?= \bigcap_{\alpha \in A} \{\pi_{\alpha}^{-1}(E_\alpha) : E_\alpha \in \mathcal{M}_\alpha\}
$$
Any comment is appreciated.

REPLY [1 votes]: It's more of a union. One could also say $$\Pi = \{A: \exists \alpha \in A: \exists E_\alpha \in \mathcal{M}_\alpha: A = \pi_{\alpha}^{-1}[E_\alpha] \}$$
which, if you like, could also be written as 
$$\Pi = \bigcup_{\alpha \in A} \pi_{\alpha}^{-1}[\mathcal{M}_\alpha] $$
where $f^{-1}[\mathcal{A}]$ (for a family of sets in $Y$ and $f:X \to Y$ a function between sets) just means $\{f^{-1}[A]: A \in \mathcal{A} \}$.
{"set_name": "stack_exchange", "score": 1, "question_id": 2688445}