Cartesian product

**TheProphet** · August 17th 2011, 12:49 PM

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

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

Is this the cartesian product in the "traditional sense"? For example, if
$E = F = \mathbb{R}$ and $\mathcal{E} = \mathcal{F} = \mathcal{B}(\mathbb{R})$ , could an element of $\mathcal{E} \times \mathcal{F}$ be something like $(-2,2) \times (-1,1)$ ?
Or even

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

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

**hatsoff** · August 18th 2011, 05:01 AM

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

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

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

**girdav** · August 18th 2011, 08:19 AM

Originally Posted by hatsoff

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

It's the smallest $\sigma$ -algebra (for the inclusion) which contains $\mathcal E\times \mathcal F$ .

**TheProphet** · August 19th 2011, 11:38 AM

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

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