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Thread: Cartesian product

  1. #1
    Junior Member TheProphet's Avatar
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    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?
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  2. #2
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    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.
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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 $\displaystyle \sigma(\mathcal{E}\times\mathcal{F})$ is supposed to mean.
    It's the smallest $\displaystyle \sigma$-algebra (for the inclusion) which contains $\displaystyle \mathcal E\times \mathcal F$.
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  4. #4
    Junior Member TheProphet's Avatar
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    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"?
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