Sufficient condition for convergence in $L^1$

Question

I’d like to know if the following is true: $f_n \to f$ pointwise and $\int f_n \to \int f$ imply $f_n \to f$ in $L^1$. It is well known that the assertion is true if we replace $\int f_n \to \int f$ with $\int |f_n| \to \int |f|$.

Would you give me any comment about this question. Thanks in advance!

Answer 1

$f_n=n(\chi_{(0,1/n)}-\chi_{(1/n,2/n)})$.