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Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $sigma$-algebra $mathscrB(B)$. Let $R:Bto mathbbR$ be a bounded linear operator.

Let $(Omega,mathcalF,P)$ be a probability space. Let $F$ be a $mathscrB(B)|mathcalF$-measurable mapping $Omegato B$. 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 $[E F](x):=E[F(x)]$)?

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You do not need the separability of $B$ to define $E F$ for a random vector $FcolonOmegato B$; 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 $omegainOmega$.

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 $E F:=lim_nE F_n$, with naturally defined $E F_n$.

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

See e.g. Yosida, Sections V.4 and V.5 for further details.

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.

A counterexample for random variable with nonseparable range.

Let $omega_1$ be the smallest uncountable ordinal. Let $Omega = [0,omega_1)$, the set of countable ordinals, with the order topology. Then the Banach space $B = C[0,omega_1)$ will be used. Note: every real-avalued continuous function on $[0,omega_1)$ is bounded. Moreover,
$$
lim_alpha to omega_1 f(alpha)
$$
exists for every $f in B$ and
$phi : B to mathbb R$ defined by
$$
phi(f) = lim_alpha to omega_1 f(alpha)
$$
is a bounded linear functional.
[The Stone-Cech compactification of $Omega$ is the one-point compactification of $Omega$.]

Our measure space is $(Omega,mathcal F, mathbb P)$, where $mathcal F$ is the countable-cocountable sigma-algebra, and $mathbb P$ is $0$ on countable sets and $1$ on cocountable sets. Note: if $f in C[0,omega_1)$, then the integral is
$$
int f(alpha);mathbb P(dalpha) = lim_alpha to omega_1 f(alpha) = phi(f) .
$$

Define $F : Omega to B$ by $F(alpha) = mathbf1_(alpha,omega_1) $, the indicator function of the interval $(alpha,omega_1)$, which is a clopen set.

We will show that that there is no $mathbb E[F] in B$ with the property $mathbb E[Rcirc F] = Rbig(mathbb E[F]big)$ for all bounded linear functionals $R : B to mathbb R$. Suppose it does exist.

Fix $xi in [0,omega_1)$. Then $f mapsto f(xi)$ is a bounded linear functional, and
$$
F(alpha)(xi) = begincases
1,quad alpha < xi
\
0,quad alpha ge xi
endcases
$$
and thus
$$
0 = lim_alpha to omega_1 F(alpha)(xi) =
int F(alpha)(xi);mathbb P(dalpha) =
left(mathbb E[F]right)(xi)
$$
This holds for all $xi$ so $mathbb E[F] = mathbf 0$, the zero element of $B$.

On the other hand $phi$ defined above is a bounded linear functional, and
$$
phi(F(alpha)) = lim_xi to omega_1F(alpha)(xi) = 1
$$
for all $alpha$.

So
$$
mathbb E[phicirc F] =intphi(F(alpha));mathbb P(dalpha)
=int 1;mathbb P(dalpha) = 1 .
$$
This is not equal to $phibig(mathbb E[F]big) = phi(mathbf 0) = 0$.

.....

Now I'm wondering if $F$ is Borel.