Suppose that $f_n\to f$ in measure and $|f_n|\le g\in L^1$, for all $n$. Show that $f_n\to f$ in $L^1$. That is $\lim_n\int_X|f_n-f|d\mu=0$.
I have already shown that $\int_Xfd\mu=\lim_n\int_X f_nd\mu$, but don't see how to use this or anything else to get to the desired result.