$\newcommand{\E}{\mathsf{E}}$ $\newcommand{\P}{\mathsf{P}}$ The following additional question was asked in a comment by user Oleg:

Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ is a (not necessarily separable) Banach space, and $F\colon\Omega\to B$ is a strongly measurable random vector in $B$, with $\E\|F\|<\infty$. If now $\mathcal G$ is a sub-σ-algebra of $\mathcal F$, is then there a well-defined and strongly measurable conditional expectation $\E(F|\mathcal G)$?

The strong measurability of a random vector $F$ means that there is a sequence of finitely-valued random vectors $F_n$ in $B$ such that $\|F_n(\omega)-F(\omega)\|\to0$ for $\P$-almost all $\omega\in\Omega$.

I thought it makes sense to post this question separately, which is what is being done here.