1. ## Cartesian product

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?

2. ## Re: Cartesian product

The "traditional" cartesian product---let's denote it $\displaystyle \times_{\text{trad}}$---is usually given by $\displaystyle A\times_{\text{trad}} B=\{(a,b):a\in A,b\in B\}$. So the cartesian product of $\displaystyle \sigma$-algebras as given in your post is not the same. Let's denote that by $\displaystyle \times_\sigma$.

So suppose $\displaystyle \Lambda\in\mathcal{E}$ and $\displaystyle \Gamma\in\mathcal{F}$. Then an element in $\displaystyle \mathcal{E}\times_\sigma\mathcal{F}$ has the form $\displaystyle \Lambda\times_{\text{trad}}\Gamma$, and an element in $\displaystyle \mathcal{E}\times_{\text{trad}}\mathcal{F}$ has the form $\displaystyle (\Lambda,\Gamma)$. However, this is no big deal since $\displaystyle \Lambda\times_{\text{trad}}\Gamma\mapsto(\Lambda, \Gamma)$ defines a bijective map.

I'm not sure what the notation $\displaystyle \sigma(\mathcal{E}\times\mathcal{F})$ is supposed to mean.

3. ## Re: Cartesian product

Originally Posted by hatsoff
I'm not sure what the notation $\displaystyle \sigma(\mathcal{E}\times\mathcal{F})$ is supposed to mean.
It's the smallest $\displaystyle \sigma$-algebra (for the inclusion) which contains $\displaystyle \mathcal E\times \mathcal F$.

4. ## Re: Cartesian product

Ok, so an element $\displaystyle A$ in $\displaystyle \mathcal{E} \times \mathcal{F}$ is of the form $\displaystyle A = \Lambda \times \Gamma$, where $\displaystyle \Lambda \times \Gamma$ is the traditional cartesian product and thus hold many "products"?