Here's what I really wanted to ask
Suppose that $f_n \rightarrow f$ a.e.
If we have $\int{\left|\lvert f_n\rvert^p - \lvert f\rvert^p\right|} \rightarrow 0$, is it true that $\|f_n\|_p \rightarrow \|f\|_p$?
What I want to do is let $g_n = \lvert f_n\rvert^p, g=\lvert f\rvert^p$ and then use the fact that $\int{\lvert g_n - g\rvert} \rightarrow 0$ if and only if $\int{\lvert g_n\rvert} \rightarrow \int{\lvert g\rvert}$. Then we get $\|f_n\|_p^p \rightarrow \|f\|_p^p$ and thus $\|f_n\|_p \rightarrow \|f\|_p$. Is this valid?
I'm kind of a beginner at this stuff.