Suppose $f_n$ is as sequence of functions in $L^1[0,1]$ such that $f_n$ converges pointwise a.e. to $f\in L^1[0,1]$. Suppose also that $\int \vert f_n\vert \rightarrow \int \vert f\vert$. Is it true that $f_n$ converges to $f$ in the $L^1$ norm?
From Javaman's comment below: $|f_n-f|\leqslant |f_n|+|f|$. So DCT applies. Since $f_n\rightarrow f$ a.e. we have $\lim_n\int |f_n-f|=0.$ i.e $\Vert f_n-f\Vert \rightarrow 0.$