Sum of two measurable functions is measurable I am trying to understand the proof of Royden when $f$ and $g$ are measurable functions, then $f+g$ is measurable.
For $x \in E$ if $f(x)+g(x) < c$, then $f(x) < c - g(x)$. By the density of the rational numbers, there exists $q \in \mathbb{Q}$ such that
$f(x) < q < c - g(x)$.
Hence, $\{x \in E : f(x) + g(x)< c\} = \displaystyle\bigcup_{q \in \mathbb{Q}} [ \{x \in E: g(x)<c-q\}\cap\{x \in E: f(x)<q\}]$.
I understand almost everything, but I don't understand why we need to use the $\bigcup$ of rational numbers. i.e., I understand that we need to use a rational number in the equation to proof each part is measurable. But why to use the union of the sets that meet the conditions for every rational number? Is not enough to meet just one national number?

"For some" is another way to say "union".
$$
\{x : x \in A_q \text{ for some }q \in \mathbb Q\}
= \bigcup_{q \in \mathbb Q} A_q .
$$
Similarly, "for all" is another way to say "intersection".
$$
\{x : x \in A_q \text{ for all }q \in \mathbb Q\}
= \bigcap_{q \in \mathbb Q} A_q .
$$
