Assume: (a) $\{f_n\} \subset L^p$, (b) $f_n \to f \text{ }\mu\text{-a.e.}$ and (c) $\|f_n\|_{L^p} \to \|f\|_{L^p}\to0$.
Then show that $$\|f_n - f \|_{L^p} \to 0$$ using Fatou's Lemma, first for $p=1$ and then for $p>1$.
I looked at the function $g_n = |f_n|+|f|-|f_n-f|$. It turns out that using Fatou's Lemma on $g_n$ gives: $$\liminf_{n} \int|f_n-f|d\mu=0$$ only, leaving the case of $\limsup_n$. I noted however, that:
$$|g_n|\leq|f_n| +|f|$$
and DCT can be applied for at least the case of $p=1$ to prove $\|f_n-f\|_{L^1}\to 0$. I suspect for $p>1$ Minkowski's inequality will become handy. Am I correct. Is there an elegant way to use Fatou's Lemma here?
