Difficulty in proving that the sum of two measurable function is a measurable function Let $(\Omega, \Sigma)$, $(\mathbb{R},\mathcal{B})$ be the two measurable space. Let $f:\Omega\rightarrow \mathbb{R}$, and $g:\Omega\rightarrow \mathbb{R}$ be the two measurable functions. In order to prove that $f+g$ is a measurable map, I need to show that
$$\{\omega\in\Omega: f(\omega)+g(\omega)< x\}\in \Sigma, \forall x\in\mathbb{R}$$
I was reading a proof, which says that:
$$\{\omega\in\Omega: f(\omega)+g(\omega)< x\}= \bigcup_{r\in\mathbb{Q}}\Big[\{\omega : f(\omega)<r\}\cap\{\omega: g(\omega)< x-r\}\Big].$$
I have no clue about how the above two quantities are equal. Can somebody simplify it?

\begin{align*}
f(\omega )+g(\omega )<x&\iff f(\omega )<x-g(\omega )\\
&\iff \exists r\in \mathbb Q:f(\omega )<r<x-g(\omega ),
\end{align*}
where the last equivalence comes from the density of $\mathbb Q$ in $\mathbb R$.
