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Do you know where to find references on the following questions (I suspect they are well-known)?

1) The product $\sigma$-algebra is defined to be the smallest $\sigma$-algebra containing all the "measurable rectangles." Is it possible to write every element in the product $\sigma$-algebra as a sequence of countable unions and intersections of the measurable rectangles? (I suspect the answer is no, but I can't think of a good reason why.)

(More precisely, given two $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$, is it possible to an arbitrarily element of $\mathcal{A}\times \mathcal{B}$ as $\bigcup_{i_{1}=1}^{\infty}\bigcap_{i_{2}=1}^{\infty}\cdots\bigcap_{i_{n}}^{\infty}A_{i_{1},\ldots,i_{n}}\times B_{i_{1},\ldots,i_{n}}$ for some $n$ and all $A_{i_{1},\ldots,i_{n}}\in\mathcal{A},\ B_{i_{1},\ldots,i_{n}}\in\mathcal{B}$?)

2) Given two independent $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$ (with respect to a measure $\mu$), why is not possible describe the measure $\mu$ on $\sigma(\mathcal{A}\cup\mathcal{B})$ as a product measure $\mathcal{A}\times \mathcal{B}$ (where we would define the measure of a rectangle $A\times B$ for $A\in\mathcal{A}$ and $B\in\mathcal{B}$ as $\mu(A\cap B)=\mu(A)\mu(B)$)?

Again, it makes sense that there are lot more ways to construct independent $\sigma$-algebras than taking products, but I feel like I don't understand a lot if I can't distinguish between the two.

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    See Asaf's answer [here](http://math.stackexchange.com/a/54179) for a "bottom-up" description of the $\sigma$-algebra generated by a set. Question 1) is about the special case of the $\sigma$-algebra generated by the measurable rectangles.2012-07-29
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    What do you mean by independent $\sigma$-algebras?2012-07-29

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