Need some help in understandig the following. If $\displaystyle \mathcal{E} $ and $\displaystyle \mathcal{F} $ are $\displaystyle \sigma$-algebras on $\displaystyle E$ and $\displaystyle F$ respectively, then

$\displaystyle \mathcal{E} \times \mathcal{F} = \{A \subset E \times F \, : \, A = \Lambda \times \Gamma, \, \Lambda \in \mathcal{E}, \, \Gamma \in \mathcal{F} \} $.

Is this the cartesian product in the "traditional sense"? For example, if

$\displaystyle E = F = \mathbb{R} $ and $\displaystyle \mathcal{E} = \mathcal{F} = \mathcal{B}(\mathbb{R}) $, could an element of $\displaystyle \mathcal{E} \times \mathcal{F}$ be something like $\displaystyle (-2,2) \times (-1,1) $ ?

Or even

$\displaystyle \left( \bigcup_{n \geq 1} A_{n} \right) \times \left( \bigcap_{n \geq 1} B_{n} \right), \; (A_{n})_{n \geq 1} \subset \mathcal{E}, \; (B_{n})_{n \geq 1} \subset \mathcal{F} $ ?

Also, since $\displaystyle \mathcal{E} \otimes \mathcal{F} = \sigma( \mathcal{E} \times \mathcal{F}) $, what does for example the intersection of cartesians products mean?