A generalized rectangle is $A\times B$ where $A$ and $B$ are subsets of $\Bbb R.$ Is the $\sigma$-algebra $S$ on $\Bbb R^2$, generated by the set of all generalized rectangles, equal to the set of all subsets of $\Bbb R^2\;$?
I suspect not. Identify $\Bbb R$ with the cardinal ordinal $2^{\omega}$ and let $T=\{(a,b): a\in b\in 2^{\omega}\}.$ I suspect that $T\not \in S.$ I worked on this but I feel that I am stuck, perhaps on something obvious.
