Is there an easy/elementary way to show that if $f_k\to f \in L^p(\Omega)$ we can conclude that $\int |f_k|^p\to \int |f|^p$

Question

Hey just stumbled upon this problem.

Often one finds the Lebesgue dominated convergence theorem presented for $L^1$ spaces with the statement that the integrals do converge, i.e. $\int f_k \to \int f$.

However, in the versions for $L^p$ spaces with $p\in(1,\infty)$ this statement is missing (at least in my references) i.e. one does not find $\int |f_k|^p\to \int |f|^p$. I found one reference that gives a positive answer: From $f_k\to f \in L^p$ one can conclude that $|f_k|^p \to |f|^p \in L^1$. However the proof I found relies on the Vitali convergence theorem. Is there a more elementary proof to conclude the convergence of the integrals?

Answer 1

Let $(X,\|\cdot\|)$ be a Normed space and assume that $x_n\to x$. By one hand, note that $$\|x_n\|=\|x_n-x+x\|\leq\|x_n-x\|+\|x\|$$

On the other hand $$\|x\|=\|x-x_n+x_n\|\leq \|x_n-x\|+\|x_n\|$$

which implies that $$\tag{1}|\|x_n\|-\|x\||\leq\|x_n-x\|$$

From $(1)$ we conclude that $\|x_n\|\to\|x\|$.

To answer Op's question, we apply the previous result by taking $X=L^p(\Omega)$ and noting that the function $x\mapsto x^p$ is continuous for $p\geq 1$.

Remark: This answer was considerably improved by @sranthrop.