Prove that $(f+g)^{-1}=\bigcup_{t+s >a, t,s \in \mathbb{Q}}{f^{-1}(t,\infty) \cap g^{-1}(s,\infty)}$

Question

Prove that $$(f+g)^{-1}=\bigcup_{t+s >a, t,s \in \mathbb{Q}}{f^{-1}(t,\infty) \cap g^{-1}(s,\infty)}$$

The equality above is used to show that the sum of two measurable functions is measurable [here] page 3.

Question: How do we obtain such equality intuitively?

Answer 1

Is it not intuitively clear that $f(x)+g(x) > a$ if and only if there exists $(t,s)$ with $f(x) > t$ and $g(x) > s$ such that $t+s > a$?