Let $\Sigma$ denotes the Borel $\sigma$-algebra of $\mathbb{R}$ and $\Delta=\{(x,y)\in\mathbb{R}^2: x=y\}$. I am trying to clarity the definitions of $\Sigma\times\Sigma$ (the sets which contains all measurable rectangles) and $\Sigma\otimes\Sigma$ (the $\sigma$-algebra generated by the collection of all measurable rectangles). My question is (1) does $\Delta$ belong to $\Sigma\times\Sigma$? (2) does $\Delta$ belong to $\Sigma\otimes\Sigma$?
I am thinking that (1) would be no (since a measurable rectangle can be arbitrary measurable sets which are not required to be intervals?) and (2) would be yes (can we write $\Delta$ like countable unions of some open intervals? I cannot find a one at time).
Best Answer
Yes. Draw this line in the plane. Now you can easily see how it is a diagnonal of squares whose side length is $1/n$. The union of these squares is in $\Sigma$; denote it by $U_n$. Notice that the diagonal line is $$\cap_n U_n.$$