If $f_n \to f$ $\mu$-a.e. and $\|f_n\|_1 \to \|f\|$, then $\|f_n - f\| \to 0$

Question

I'm trying to prove this below result

Theorem: Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Let $f_n, f \in \mathcal L_1 (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e. and $\|f_n\|_1 \to \|f\|$ as $n \to \infty$. Then $\|f_n - f\| \to 0$.

Could you have a check if I correctly apply Fatou's lemma.

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My attempt: We have $$ \begin{align} \limsup_n \int |f_n-f| &= - \liminf_n \int -|f_n-f|\\ &= - \liminf_n \int [|f_n| + |f|-|f_n-f|-(|f_n|+|f|)] \\ &\le - \liminf_n \int (|f_n|+|f|-|f_n-f|) - \liminf_n \int-(|f_n| +|f|) \\ &= - \liminf_n \int (|f_n|+|f|-|f_n-f|) + \limsup_n \int(|f_n|+|f|). \end{align} $$

• Because $f_n \to f$ $\mu$-a.e., we get $|f_n|+|f|-|f_n-f| \to 2|f|$ $\mu$-a.e. By Fatou's lemma, $$ \liminf_n \int (|f_n|+|f|-|f_n-f|) \ge 2\int |f| = 2 \|f\|_1. $$
• Because $\|f_n\|_1 \to \|f\|_1$, we get $$ \limsup_n \int (|f_n| + |f|) = \|f\|_1 + \limsup_n \|f_n\|_1= 2 \|f\|_1. $$

It follows that $$ \lim_n \|f_n-f\|_1 \le \limsup_n \|f_n-f\|_1 = \limsup_n \int |f_n-f| \le 0. $$

This completes the proof.

Answer 1

I have found that this result can be generalized to $f, f_n \in\mathcal L_p (X, \mu, E)$, i.e.,

Theorem: Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space, $(E, |\cdot|)$ a Banach space, and $p \in [1, \infty)$. Let $f_n, f \in \mathcal L_p (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e. and $\|f_n\|_p \to \|f\|_p$ as $n \to \infty$. Then $\|f_n - f\|_p \to 0$.

Proof:

First, we have $|x-y|^p \le (|x|+|y|)^p \le (2 \max \{|x|, |y|\})^p = 2^p (\max \{|x|, |y|\})^p \le 2^p (|x|+|y|)^p$ for all $x,y \in E$. It follows that $$ 2^p (|f_n|^p + |f|^p) - |f_n-f|^p \ge 0 \quad \forall n. $$

By Fatou's lemma, we get $$ \int \liminf_n [2^p (|f_n|^p + |f|^p) - |f_n-f|^p] \le \liminf_n \int [2^p (|f_n|^p + |f|^p) - |f_n-f|^p]. $$

• Notice that $f_n \to f$ $\mu$-a.e. implies $2^p (|f_n|^p + |f|^p) - |f_n-f|^p \to 2^{p+1} |f|^p$ $\mu$-a.e.

• If $a_n, b_n \in \mathbb R$ such that that $(a_n)$ is convergent in $\mathbb R$, then $\liminf (a_n+b_n) = \lim a_n + \liminf b_n$. Then $$ \begin{align} \liminf_n \int [2^p (|f_n|^p + |f|^p) - |f_n-f|^p] &=\lim_n \int 2^p (|f_n|^p + |f|^p) +\liminf_n \int - |f_n-f|^p \\ &=2^p \lim_n \int (|f_n|^p + |f|^p) - \limsup_n \int |f_n-f|^p. \end{align} $$

It follows that $$ \int 2^{p+1} |f|^p \le \int 2^{p+1} |f|^p - \limsup_n \int |f_n-f|^p. $$

This completes the proof.