In generalizing Fubini's theorem to functions on Banach space, I need to use below convergence result. Could you check if it's correct?
Related definitions of Bochner integrals can be found here. Let
$(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, | \cdot |)$ a Banach space.
$\mathcal S (X, \mu, E)$ the space of $\lambda$-simple functions from $X$ to $E$.
$\mathcal L_0 (X, \mu, E)$ the space of $\mu$-measurable functions from $X$ to $E$.
$\mathcal L_1 (X, \mu, E)$ the space of $\mu$-integrable functions from $X$ to $E$.
Theorem: Let $(f_n)$ be a sequence in $\mathcal L_1 (X, \mu, E)$ that converges to $f$ in $\mathcal L_1 (X, \mu, E)$. Then there exists a subsequence $(f_{\varphi(n)})$ that converges $\mu$-a.e. to $f$.
My attempt: It suffices to assume $f =0$, because, if $f \neq 0$, we can consider the sequence $(f_n - f)$.
Every convergent sequence in metric spaces (or spaces endowed with semi-norm such as $\mathcal L_1 (X, \mu, E)$) is a Cauchy sequence. It follows that $(f_n)$ is a Cauchy sequence in $\mathcal L_1 (X, \mu, E)$. Then there is a subsequence $\varphi$ of $(n)$ such that $\|f_m - f_n\|_1 \le 2^{-2\ell}$ for $m,n \ge \varphi(\ell)$. Let $h_\ell :=f_{\varphi(\ell)}$. Then $\|h_\ell - h_m\|_1 \le 2^{-2\ell}$ for $m \ge \ell$. Take the limit $m \to \infty$, we get $\|h_\ell\|_1 \le 2^{-2\ell}$.
Let $B_\ell := \{ x\in X \mid |h_\ell (x)| \ge 2^{-\ell} \}, A_n := \cup_{\ell \ge n} B_\ell$, and $A := \cap_n A_n$. Then $\mu(B_\ell) \le 2^{-\ell}$ for $\ell \in \mathbb N$, $\mu(A_n) \le 2^{-n+1}$ for $n \in \mathbb N$, and $\mu(A) =0$. Then $(h_\ell)$ converges to $0$ uniformly on each $A_n^c$ and pointwise on $A^c$. This completes the proof.
