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Math Help - Cartesian product

  1. #1
    Junior Member TheProphet's Avatar
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    Cartesian product

    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?
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  2. #2
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    Re: Cartesian product

    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.
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  3. #3
    Super Member girdav's Avatar
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    Re: Cartesian product

    Quote Originally Posted by hatsoff View Post
    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.
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  4. #4
    Junior Member TheProphet's Avatar
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    Re: Cartesian product

    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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