Let ($\Omega, \mathcal{F}, P$) be a probability space. Definition of Ito's integral to processes in $\mathcal{V}$ (Oksendal book): Let $f \in \mathcal{V}$, then $f:[0,\infty)\times \Omega \to \mathbb{R}$ is $\mathcal{B}[0, \infty]\otimes \mathcal{F}$-measurable, is adapted w.r.t. brownian motion and $f \in L^2([0, \infty)\times \Omega)$. With these conditions is possible define a Ito integral. The book of Oksendal gives a first extension: Change "adapted w.r.t. brownian motion" with "adapted w.r.t. a filtration $(\mathcal{H}_t)_{t \geq 0}$, and the brownian motion is a martingale w.r.t this filtration". I can define with the same way the Ito integral: Using elementary process $\phi_n = \sum \eta_j 1_{[t_{j-1}-t_j)}$ and $$\phi_n \to f \hspace{1mm} \text{ in } L^2([0, \infty)\times \Omega).$$
But I need the Ito isometry in elementary processes to get uniqueness. I can't prove that, because in this context $\eta_j$ is measurable w.r.t $\mathcal{H}_{t_j}$ insted of $B_{t_j}$. In the initial case the Ito isometry for elementary process is $$E\left( \int^T_S\phi(t,\omega)dB_t(\omega) \right)^2 = E\left[ \int^T_S\phi(t,\omega)^2dt \right]$$ It is proved in this way: If $i \neq j$ by independence of brownian motion $$E[\eta_i \eta_j \Delta B_i \Delta B_j] = 0$$ Then, when $i=j$ $$E[\eta_i \eta_j \Delta B_i \Delta B_j] = E[\eta^2_i](t_{j+1}-t_j).$$ In the other case I can't use independence. Even I can't prove the case $i=j$. Thanks for any hint or maybe in this case this is not the way to prove uniqueness.