This is problem 4T in Bartle's The elements of integration and Lebesgue measure.
Suppose $f_n$ are non-negative measurable function such that $(f_n)$ converges to $f$, and that $$\int fd\mu = \lim \int f_n d\mu<\infty.$$ Prove that for all measurable set $E$, we have $$\int_E fd\mu = \lim \int_E f_n d\mu.$$
I found a solution here.
The solution used the reverse Fatou's lemma, which needs the hypothesis of dominated boundedness, that is there exists a measurable function $g$ such that $f_n\le g$ for all $n$ and $\displaystyle \int g <\infty$. I cannot see how can we get this function $g$.
Thank you very much.