The proofs for , "if $f,g :X\to \mathbb{R}$ are integrable then $f+g$ is (Lebesgue) measurable" is available in any Real Analysis textbook.
$\textbf{BUT}$ I have a doubt about it and also the proof if we assume that $f:X\to \mathbb{R}$ and $g :Y\to \mathbb{R}$. Does it make any difference?
My guess is no and still we can write the proof for $f(x)+g(y)$ is measurable iff the following set is (Lebesgue) measurable: (for a fixed $t$) $$\{(x,y)\in X \times Y\ s.t. f(x)+g(y)> t \} = \bigcup_{r,q\in \mathbb{Q} \\ r+q > t} \big[\{(x,y)\in X\times Y | f(x)>q \}\cap \{(x,y)\in X \times Y | g(y)>r \} \big]$$
Is above true?