Bochner Integral: Approximability Problem
Given a measure space $\Omega$ and a Banach space $E$.
Consider a Bochner measurable function $S_n\to F$.
Then it admits an approximation from nearly below:
$$\|S_n(\omega)\|\leq \vartheta\|F(\omega)\|:\quad S_n\to F\quad(\vartheta>1)$$
(This is sufficient for most cases regarding proofs.)
Can it happen that it does not admit an approximation from below:
$$\|S_n(\omega)\|\leq\|F(\omega)\|:\quad S_n\to F$$
(I'm just being curious wether it can actually fail.)
Constructions
The only constructions I found so far are cutoff to bound an approximation:
$$E_n:=\{\|F_n\|\leq2\|F\|\}:\quad F'_n:=F_n\chi_{E_n}\implies\|F'_n\|\leq2\|F\|$$
and reset to obtain an increasing resp. decreasing approximation:
$$R^\pm_n:=\{\|F_n\|\gtrless\|F'_{n-1}\|\}:\quad F'_n:=F_n\chi_{\Omega_n}+F'_{n-1}\chi_{\Omega_n^\complement}\implies\|F_n'\|\updownarrow\|F'_{n+1}\|$$
Apart from these truncation seems less relevant here:
$$\Omega_n\uparrow\Omega:\quad F'_n:=\chi_{\Omega_n}\implies\|F'_n\|\uparrow\|F\|$$

This is a modified version from: Cohn: Measure Theory
Enumerate a countable dense set:
$$\#S\leq\mathfrak{n}:\quad S=\{s_1,\ldots\}\quad(\overline{S}=F\Omega)$$
Regard the finite subsets:
$$S_K:=\{s_1,\ldots,s_K\}$$
Construct the domains by:
$$A_k:=A_n(s_k):=\{\omega:\|s_k\|\leq\|F(\omega)\|\}\cap\{\omega:\|F(\omega)-s_k\|<\tfrac{1}{n}\}$$
And sum up their disjoint parts:
$$A_k':=A_k\setminus\left(\bigcup_{l=1}^{k-1}A_l\right):\quad F_n:=\sum_{k=1}^{K=n}s_k\chi_{A_k'}$$
(Note that the supports are clearly measurable.)
