Show that a set is measurable ( Lecture Jean-François Le Gall )

Question

I am reading the demonstration of the monotone convergence theorem in the tuition in French by Jean-François Le Gall. To show the inequality

$ \int f d\mu \leq \lim_{x \to \infty} \uparrow \int f_{n} d\mu $

a staged positive function is chosen :

$ h = \sum_{i=1}^{m} \alpha_{i} 1_{A_{i}}$ with $ h \leq f$.

Then choosing $a \in [0;1[$ and defining $E_{n} = \{x \in E : ah(x) \leq f_{n}(x) \}$ :

it is said that $E_{n}$ is measurable. How to properly prove this fact ?

Answer 1

Since $h$ and $f_n$ are measurable, the function $g(x):=ah(x)-f_n(x)$ is measurable, and thus $$E_n=g^{-1}((-\infty ,0])\cap E$$ is measurable (if $E$ is measurable).

Answer 2

Assuming that the functions are $\mathbb R$ valued we can write $E_n=E\cap [(f_n-ah)^{-1} ([0,\infty)]$ and $f_n-ah$ is measurable.

In case infinite values are allowed you can write $E_n=[\cup_{q \in \mathbb Q} (ah)^{-1} ((q, \infty) \cap f_n^{-1} (-\infty,q)]^{c}$