Have you any mind for the case that $f$ and $z$ depend on both $B$ and $X$ (I mean to the elements of these spaces)?
If not can we put conditions on $z$? for example $z$ becomes sufficiently small.
As I deal with this problem, the answer of my first question is NO, for the second question, I write: We define $s := \max_{\xi \in B} \{z (\xi , x)\}$, by Fubini's theorem we have $\int_{B}\int_{D} g (z - s) dx d\mu(\xi) = \int_{B\times D} g (z - s) dx d\mu(\xi)$ $\leq \|(z - s)\| \|g\|$ $\leq 2 \|s\| \|g\|$ as $\|z\| \rightarrow 0$ thus $\|s\| \rightarrow 0$, now one can write $\int_{B \times X} g z dx d\mu(\xi) = \int_{B \times X} g s dx d\mu(\xi) $ $= \int_{X} s \int_{B} g d\mu(\xi) dx = \langle E[g] ,s \rangle,$ also noting that
$\langle E[g] ,E[z] - s \rangle \leq \int_{D}\int_{B} (z - s) d\mu(\xi)dx \int_{D} \int_{B} g d\mu(\xi)dx $
Am I right? In advance thanks for your contribution.
And one more, norms belongs to proper spaces and are not same.
If you have any question I am eager to here.