If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

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Let $1\leq p < \infty$. Suppose that

  1. $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite),
  2. $f_k \to f$ almost everywhere, and
  3. $\|f_k\|_{L^p} \to \|f\|_{L^p}$.

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

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