Why is $g_n:=\inf\{f_n,f_{n+1},...\}$ integrable if $f_n$ are?

Question

This question is motivated by the proof of the Fatou's lemma.

My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise minimum of any two integrable functions is integrable. How can this be extended to this case?

Update:

After the two answers, it turned out that our ways of introducing the integration theory is not very conventional: we haven't had Borel sets yet.

Could some manipulation with minimum of finite amount do the job?

Say, we define $g_{n,m}:=\min\{f_n,\ldots,f_m\}$. Then $g_{n,m}\searrow g_n$ for $m \to \infty$ and we're done with the monotone convergence theorem. Would it work?

Answer 1

Note that, for every $a\in\mathbb R$ $$ g_n^{-1}(-\infty,a)=\bigcup_{m\ge n}f_m^{-1}(-\infty,a). $$ The right hand side is a measurable set, as a countable union of measurable sets, while the left hand side, spans, for different values of $a$ all the inverses of Borel sets by $g_n$. Hence, all the $g_n$'s are measurable.