I'm trying to follow Bartle's proof of the Monotone Convergence Theorem in measure theory.
If $f_n$ is an increasing sequence of measurable functions converging to $f$, then $$\lim \int f_n \,d\mu = \int f \, d \mu.$$
The first step of the proof is to pick a real number $0 < \alpha < 1$ and a simple function $0 \leq \varphi \leq f $ and define the set $$A_n = \{ x \in X : f_n(x) \geq \alpha \varphi(x) \} .$$ My question is: why is this set measurable?
Since $f_n$ is measurable, the preimage $f_n^{-1}(\mathbb{R}_{> a})$ is measurable for all real numbers $a$. But in this case $\alpha \varphi(x)$ depends on $x$, so I don't know how to proceed. I appreciate any help.
