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.