Measurability of $[f<g]$ where $f$, $g$ are $\mathbb{R}\cup\{\pm\infty\}$-valued

Question

Let $(X,\mu,\mathcal{A})$ be a measure space, and let $f,g\in\mathcal{L}_0(X,\mu,\bar{\mathbb{R}})$, that is let $f$ and $g$ be $\mathbb{R}\cup\{\pm\infty\}$-valued and $$[f<\alpha]:=\{x\in X\colon f(x)<\alpha\},[g<\alpha]\in\mathcal{A},\quad \alpha\in\mathbb{R}.$$

I want to show that $[f<g]\in\mathcal{A}$. How? I tried to write $[f<g]$ as a countable intersection/union of sets of the above form but I did not succeed.

Answer 1

$$[f<g]=\bigcup_{q\in\mathbb Q}([f<q]\cap[q<g])$$