For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] , \delta s \rangle_{V^{*},V} $$ Why this relation is true?
$D \subsetneq \mathbb{R}$ and $T \in \mathbb{R}$ is a real number.
A similar question was asked in Exchangeability of inner product with the integral by @jpv, I have extended the question and propose a proof for that, but I am not completely sure about that proof.