show that $\|f_n-f\|_p \to 0 \iff \|f_n\|_p \to \|f\|_p$

Question

Suppose that $1 \le p \lt \infty$. If $f_n,f \in L^p$ and $f_n \to f$ a.e, then show that $\|f_n-f\|_p \to 0 \iff \|f_n\|_p \to \|f\|_p$

Hint: If $f_n,g_n,f,g \in L^1, f_n \to f \text{ and } g_n \to g \text{ a.e }. |f_n| \le g_n \text{ and } \int g_n \to \int g$, then $\int f_n \to \int f$.

Proof (hint): Apply Fatou's Lemma to $g_n \pm\text{Re}({f_n})$ and $g_n\pm\text{Im}({f_n})$ respectively to get the result.

Now ($\implies $) follows by Triangle inequality. For ($\impliedby )$ we have $$|f_n-f|^p \le (|f_n|+|f|)^p \to 2^p |f|^p$$ Moreover $$\left(\int\left(|f_n|+|f| \right)^p \right)^{\frac{1}{p}} \le \|f_n\|_p+\|f\|_p \to 2\|f\|_p$$ $$\implies \limsup_n\left(\int\left(|f_n|+|f| \right)^p \right)^{\frac{1}{p}} \le 2\|f\|_p$$ By Fatou's Lemma we have $$\int \liminf_n \left(|f_n|+|f|\right)^p \le \liminf_n \int \left(|f_n|+|f|\right)^p $$ $$\implies 2^p \int |f|^p \le \liminf_n \int \left(|f_n|+|f|\right)^p \le \limsup_n \int \left(|f_n|+|f|\right)^p \le 2^p \|f\|_p^p$$

This gives us that $$\lim_n \int \left(|f_n|+|f|\right)^p=\int 2^p |f|^p=2^p||f||_p^p$$ Then $h_n=|f_n-f|^p \to 0$ and $|h_n| \le g_n=(|f_n|+|f|)^p \to 2^p|f|^p=g$. Also $\int g_n \to \int g$. Thus by the hint we have $\int |f_n-f|^p\to 0$.

This seems alright to me. I just wanted to confirm. Thanks for the help!

Answer 1

This is similar to exercise 10 in Folland.

Exercise 10

Suppose $1\leq p < \infty$. If $f_n,f\in L^p$ and $f_n\rightarrow f$ a.e., then $\|f_n - f\|_{p}\rightarrow 0$ if and only if $\|f_n\|_{p}\rightarrow \|f\|_{p}$

Proof - Let $\{f_n\}$ be a sequence in $L^p$ that converges to $f$ a.e. Suppose first that $\|f_n - f\|_{p}\to 0$ then from Minkoqski's Inequality it is a direct consequence that $$|\|f_n\|_{p} - \|f\|_{p}|\leq \|f_n - f\|_{p}$$ This implies that $$\|f_n\|_{p}\to \|f\|_{p}$$ Conversely, suppose $\|f_n\|_{p}\to \|f\|_{p}$. Recall that $|f_n - f|^p \leq 2^p(|f_n|^p + |f|^p)$. So, \begin{align*} \lim_{n\to \infty}\int |f_n - f|^{p}d\mu = \lim_{n\to \infty}\|f_n - f\|^{p}_{p}&\leq \lim_{n\to \infty}\int 2^p(|f_n|^p + |f|^p)d\mu\\ &= 2^p \lim_{n\to \infty}(\|f_n\|^{p}_{p} + \|f\|_{p}^{p})\\ &= 2^p\|f\|_{p}^{p} + 2^p \|f\|^{p}_{p}\\ &= \int 2^{p+1} |f|^p d\mu \end{align*} Therefore by the Generalized Dominated Convergence Theorem, $$\|f_n - f\|_{p} \to 0$$