(Note: All of the below are from Oksendal's Stochastic Differential Equations, chapter 3.)
Claim: For $f\in\mathcal V(0,T)$ and $0\le S<T$, we have $E\int_S^T fdBt=0$.
My attempt: I have shown (as suggested) that the claim holds if $\phi\in\mathcal V$ and $\phi$ is elementary, i.e. $\phi(t,\omega)=\sum_j e_j(\omega)1_{[t_j,t_{j+1})}(t)$ (we are given some partition $\{t_0,t_1,...,t_n\}$ of $[0,T]$), where $e_j\in\mathcal F_{t_j}$ for each $j$. Since $f\in \mathcal V$, there exists a sequence $\{\phi_n\}\subset \mathcal V$ of elementary functions so that $E[\int_S^T (f-\phi_n)^2\,dt]\to0$ as $n\to\infty$. By the isometry, we have $E(\int_S^T (f-\phi_n)\,dBt)^2\to0$. But I'm not sure how to connect the elementary result to the one we want. Would greatly appreciate some help doing so. Thanks in advance!
Definition: $f\in \mathcal V(0,T)$ means $f:[0,\infty)\times\Omega\to\mathbb{R}$ satisfies
(i) $(t,\omega)\mapsto f(t,\omega)$ is $\mathcal B\times\mathcal F$-measurable,
(ii) $f(t,\omega)\in\mathcal F_t$ for each $t\in[0,T]$, and
(iii) $E[\int_S^T f(t,\omega)^2dt]<\infty$.
Recall that $(EX)^2\le E(X^2)$, and so
$$\left(E\int_S^TfdB_t-E\int_S^T\phi_ndB_t\right)^2\le E\left(\int_S^T(f-\phi_n)dB_t\right)^2$$
Now use what you know about elementary functions and your convergence result from the Ito isometry.