Let $S = \mathbb{R}^2 \setminus \{ (x,y): x+y \in \mathbb{Q} \}.$ Show that there are no Lebesgue measurable sets $A, B \subset \mathbb{R}$ of positive Lebesgue measure for which $A \times B \subset S$.
I don't really know what I'm given and what to work with. It's obvious that $A, B$ cannot have any rational numbers. But I'm not sure how to start. Could someone help me with this?
The main result here is that if $A,B$ have positive measure then $A+B$ contains a non trivial interval (see https://math.stackexchange.com/a/1276487/27978 for a proof using convolution).
Consider the map $T((x,y)) = x+y$ and note that $(x,y) \in S$ iff $T((x,y)) \notin \mathbb{Q}$.
The above result shows that $T(A \times B) = A+B$ contains a non trivial interval, in particular, there is some $(a,b) \in A \times B$ such that $T((a,b)) \in \mathbb{Q}$ and so $(a,b) \notin S$.