$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear operator. 


Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $F$ be a $\mathscr{B}(B)|\mathcal{F}$-measurable mapping $\Omega\to B$. Suppose that $\E \|F\|<\infty$.


Question: is it true without any additional assumptions that $\E F$ is well-defined, belongs to $B$ and
$$
\E RF= R \E F?
$$


Remark: usually (e.g. in the Ledoux-Talagrand book) the separability of the space B is additionally imposed. I wonder whether the statement is true without this assumption. For example, what happens if $B$ is just a space of bounded measurable functions (in that case one can just define $[\E F](x):=\E[F(x)]$)?



Basic properties of expectation in non-separable Banach spaces
$\newcommand{\E}{\operatorname{\mathsf{E}}}$
You do not need the separability of $B$ to define $\E F$ for a random vector $F\colon\Omega\to B$; however, you need to assume that $F$ is strongly measurable, in the sense that there is a sequence of finitely-valued random vectors $F\_n$ in $B$ such that $\|F\_n(\omega)-F(\omega)\|\to0$ for almost all $\omega\in\Omega$. 


By Bochner's theorem, if $F$ is strongly measurable, then $\E\|F\|<\infty$ iff $F$ is Bochner-integrable, in the sense that for some sequence of finitely-valued random vectors $F\_n$ in $B$ we have $\|F\_n(\omega)-F(\omega)\|\to0$ for almost all $\omega$ and $\E\|F\_n-F\|\to0$; then $\E F:=\lim\_n\E F\_n$, with naturally defined $\E F\_n$. 


It is then known and easy to check that $\E RF=R\E F$ for any strongly measurable random vector $F$ with $\E\|F\|<\infty$ and any bounded linear operator $R\colon B\to\mathbb R$; that is, the Bochner integrability implies the Pettis integrability. 


See e.g. [Yosida, Sections V.4 and V.5](https://www.springer.com/gp/book/9783642962080) for further details. 




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Concerning your remark that usually the Banach space is assumed to be separable: this is done to ensure the measurability in a number of instances, including the measurability of the sum of random vectors. 


