Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra?
Solution 1:
think of the union of two rectangles in $\mathbb{R}^2$, like $[0,1]\times[0,1]\cup[2,3]\times[2,3]$. is this a product?
Solution 2:
Simple example: $A$ and $B$ are both $\mathbb{R}$ and $\mathcal{A}$ and $\mathcal B$ are both Borel $\sigma$-algebra. Then sets like $(-\infty,a)\times(-\infty,b)$ are in $\mathcal A\times \mathcal B$, but not their complements.