TITLE: why $g_n$ is measurable in the proof of Fatou's Lemma
QUESTION [5 upvotes]: Fatou Lemma: Suppose $\{f_n\}$ is a sequence of measurable functions with $f_n \geq 0$. If $\lim_{n\rightarrow\infty}f_n(x)=f(x)$ for a.e. $x$, then
$$\int f \leq \liminf_{n\rightarrow\infty}\int f_n$$
Proof: Suppose $0\leq g \leq f$, where $g$ is bounded and supported on a set $E$ of finite measure. If we set $g_n(x)=\min(g(x),f_n(x))$, then $g_n$ is measurable.
etc..

I don't know why $g_n$ is measurable. because the author didn't assume $g$ is measurable. can anyone explain why? thanks very much

REPLY [3 votes]: $g$ is assumed to be measurable, but the author didn't explicitly say so.