Is the diagonal set a measurable rectangle? 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).

Let $x\neq y$. If there is a measurable rectangle containing $(x,x)$ and $(y,y)$, there must be a set $A$ containing both $x$ and $y$ and a set $B$ doing the same such that $(x,x)$ and $(y,y)$ are in $A\times B$. But then $(x,y)\in A\times B$.
