Two questions on product measures 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.
 A: Consider the Cantor Set. It is compact, and therefore measurable.  Let $\mathcal{A}$ denote the half-open (right closed, left open) intervals, along with the null set.  Then $\mathcal{A}$ is an algebra of sets. $\sigma(\mathcal{A})$ is the Borel sets.
However, the Cantor set is totally disconnected and uncountable.  It is not a countable intersection or union of elements of $\mathcal{A}$.  The appearance of $\sigma(\mathcal{A})$ in terms of $\mathcal{A}$ is subtle.  
Consider a Cantor Set constructed along  the line $y=x$ in $[0,1]^2$.
A: 1) Certainly not be a finite sequence. You can think of $\sigma(\mathcal{A})$ as being the closure of $\mathcal{A}$ under taking countable unions and countable intersections. To see this, denote this closure by $C(\mathcal{A})$. This definitely forms a $\sigma$-algebra, and hence by definition $\sigma(\mathcal{A}) \subseteq C(\mathcal{A})$; but any $\sigma$-algebra containing $\mathcal{A}$ must contain all sets obtained by countable unions and intersections of elements of $\mathcal{A}$, and so $C(\mathcal{A}) \subseteq \sigma(\mathcal{A})$.
2) What makes you think that $\sigma(\mathcal{A} \cup \mathcal{B})$ is generated by rectangles?
