If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ Let $1\leq p < \infty$. Suppose that 


*

*$\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), 

*$f_k \to f$ almost everywhere, and 

*$\|f_k\|_{L^p} \to \|f\|_{L^p}$. 


Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$ 
A statement in the other direction (i.e. $\|f_k - f\|_{L^p} \to 0 \Rightarrow \|f_k\|_{L^p} \to \|f\|_{L^p}$ ) follows pretty easily and is the one that I've seen most of the time. I'm not how to show the result above though. 

Consider $g_k = 2^p(|f_k|^p + |f|^p) - |f_k - f|^p$.
Since $g_k \geq 0$ (why?), and $g_k \to 2^{p+1}|f|^p$ a.e., we can apply Fatou's Lemma:
$$\int \liminf g_k \leq \liminf \int g_k$$
so that
$$\int 2^{p+1}|f|^p \leq \liminf \left(\int 2^p |f_k|^p + \int 2^p |f|^p - \int |f_k - f|^p \right),$$
and I'll let you take it from here.
