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.
Answer
This is a theorem by Riesz.
Observe that
$$|f_k – f|^p \leq 2^p (|f_k|^p + |f|^p),$$
Now we can apply Fatou’s lemma to
$$2^p (|f_k|^p + |f|^p) – |f_k – f|^p \geq 0.$$
If you look well enough you will notice that this implies that
$$\limsup_{k \to \infty} \int |f_k – f|^p \, d\mu = 0.$$
Hence you can conclude the same for the normal limit.
Attribution
Source : Link , Question Author : user1736 , Answer Author : JT_NL