# Math Help - showing convergence in lebesgue integral

1. ## showing convergence in lebesgue integral

hi all
i need help in doing this problem,

Let $f_{n}, f$ be integrable functions and $f_{n} \to f$ a.e.
show that $\int |f_{n} - f| \to 0$ if and only if $\int |f_{n}| \to \int |f|$

i don't know where should i start.
any hint would be greatly appreciated

2. this is a problem from the stanford analysis textbook.
(1) $\bigl|\left|f_n\right|-\left|f\right|\bigr|\leq \left|f_n-f\right|$
gives the only if part.
(2)as for the if part, Hint: A={x:f(x)>0}, B={x:f(x)<0}, C={x:f(x)=0}.

3. For the "if" part, I would use Fatou's lemma on the sequence of positive functions $|f| + |f_n| - |f-f_n|$.