I'd like to show that for an integrable sequence of functions $f_n:X \rightarrow [0, \infty)$ with $\sup_{n\geq 1} \int_{X} f_n d\mu < \infty, f_n \rightarrow f$ pointwise a.e. for some function $f:X \rightarrow [0,\infty)$ then the claim is that:
$\int_X(f_n) \rightarrow\int_X(f)$ iff $\int_X(|f_n-f|) \rightarrow 0$
I'd like to show the direction $"\Rightarrow"$ and was told that I would probably need Fatou's lemma and the inequality $|x - y| \leq |x| + |y|$, but I'm stuck. Could somebody please give me a hint?
What I've done:
I've said that $\int_X(f_n) \rightarrow\int_X(f) \Rightarrow \lim(\int_X f_n d\mu - \int_X f d\mu) =\lim (\int_X f_n -f d\mu) = 0$ and also that $\lim( \int |f_n -f| d\mu) \geq \int \lim(|f_n -f|) d\mu$ but I feel like this is the wrong direction for what I want to show.
Then I tried the approach $\lim \int |f_n - f| d\mu \leq \lim \int f_n d\mu+ \int f d\mu$ but didn't really know what to do with that :)
Any help would be greatly appreciated! Thanks!