TITLE: Are open sets in $\mathbb{R}^2$ countable unions of disjoint open rectangles?
QUESTION [5 upvotes]: I'm just trying to prove that each open set in $\mathbb{R}^2$ is measurable using the product measure derived from the Lebesgue measure.  This is the first thing that came to mind because I recall that open sets are countable unions of disjoint open intervals, but I was wondering if there is an elementary way to do this.

REPLY [6 votes]: The answer to the question is no: the set of points lying below the line $y=-x$, for instance, is not a union of countable many pairwise disjoint open rectangles.