Assume $f_n , f \in L^1 $ and almost every where we have $ f_n \to f$ then I want to show that $\int|f_n-f| \to 0$ iff $\int|f_n| \to \int|f|$
One side is abvious by trinagle inequality , for the other side I think I must use a theorem like Dominated covergence theorem but I can not reach a conclusion. Is there a modiffied form of dominated covergence theorem.
I need some hint , thanks.