Let $X=Y=[0,1]$ and let $\mathscr M$ be the Lebesgue $\sigma$-algebra on $[0,1]$. Show that any open subset of $X\times Y$ is $\mathscr M\times \mathscr M$ measurable.
My approach: By the compactness of $X\times Y$ of every open set in $X\times Y$ will be covered by a countable disjoint collection of measurable rectangles whose union is a measurable rectangle. But I do not know what to do from there. Can someone help me out.