Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Let $p, q \in [1, \infty)$. Here we use Bochner integral. Then we have
Lemma 1 Let $f,f_n \in \mathcal L_p (X, \mu, E)$ and $g,g_n \in \mathcal L_q (X, \mu, E)$. We assume that
- $f_n \to f$ a.e. and $g_n \to g$ a.e.,
- $|f_n|^p \le |g_n|^q$ a.e. for all $n$, and $\|g_n\|_q \xrightarrow{n \to \infty} \|g\|_q$.
Then $\|f_n -f\|_p \xrightarrow{n \to \infty} 0$.
By below Lemma 2, the hypotheses of Lemma 1 imply $\|g_n - g\|_q \xrightarrow{n \to \infty} 0$.
Lemma 2 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 $\|f_n - f\|_p \to 0$.
Now I would like to generalize Lemma 1, i.e.,
Theorem Let $f,f_n \in \mathcal L_p (X, \mu, E)$ and $g,g_n \in \mathcal L_q (X, \mu, E)$. We assume that
- (H1) each sub-sequence of $(f_n)$ has a further sub-sequence that converges a.e. to $f$,
- (H2) $|f_n|^p \le |g_n|^q$ a.e. for all $n$, and $\|g_n - g\|_q \xrightarrow{n \to \infty} 0$.
Then $\|f_n -f\|_p \xrightarrow{n \to \infty} 0$.
Could you have a check if I made some subtle mistakes that I could not recognize?
My attempt Let $\varphi$ be a sub-sequence of $\mathbb N$. Let $h_n := f_{\varphi (n)}$.
Urysohn principle A sequence $(x_n)$ converges to $x$ if and only if each sub-sequence of $(x_n)$ has a further sub-sequence which converges to $x$.
By Urysohn principle, it suffices to prove that $(h_n)$ has a sub-sequence that converges to $f$ in $L^p$. By (H1), there is a sub-sequence $\psi$ of $\mathbb N$ such that $h_{\psi (n)} \xrightarrow{n \to \infty} f$ a.e. Let $l_n := g_{\varphi (n)}$. Clearly, $l_n \xrightarrow{n \to \infty} g$ in $L^q$, so there is a sub-sequence $\psi'$ of $\mathbb N$ such that $l_{\psi' (n)} \xrightarrow{n \to \infty} g$ a.e. WLOG, we assume $\psi = \psi'$. By (H2), $(h_{\psi (n)}, l_{\psi(n)})_n$ satisfies the hypotheses of Lemma 1. Hence $h_{\psi (n)} \xrightarrow{n \to \infty} f$ in $L^p$. This completes the proof.
