I am trying to prove the following extension of the Ito isometry from the elementary case to arbitrary functions in $f\in \mathcal{V}(S,T)$ where $\mathcal{V}(S,T)$ is the space of functions such that
- $(t,\omega)\to f(t,\omega)$ is $\mathcal{B}([0,\infty))\otimes \mathcal{F}$
- $f(t,\cdot)$ is $\mathcal{F}_t$-adapted for all $t\geq 0$
- $\mathbb{E}[\int_S^T f^{2}(t,\omega)dt]<\infty$
We already know that if $\{\phi_n\}$ are elementary functions then $\mathbb{E}[(\int_S^T \phi_n(t,\omega)dB_t(\omega))^2]=\mathbb{E}[\int_S^T \phi_n^{2}(t,\omega)dt]$. The idea of what I probably need to do is $\{\phi_n\}$ be elementary functions such that $\int_S^T f(t,\omega) dB_t(\omega) = \lim_{n\to\infty}\int_S^T \phi_n(t,\omega)dB_t(\omega)$ in $L^2(P)$, then I am probably trying to do the following $$\begin{align*}\mathbb{E}\Big[\Big(\int_S^T f(t,\omega) d B_t(\omega)\Big)^2\Big] &= \lim_{n\to\infty}\mathbb{E}\Big[\Big(\int_S^T \phi_n(t,\omega) d B_t(\omega)\Big)^2\Big] \\&= \lim_{n\to\infty}\mathbb{E}\Big[\int_S^T \phi_n^2(t,\omega) dt\Big]\\ &= \mathbb{E}\Big[\int_S^T f^2(t,\omega)dt\Big]\end{align*}$$
But I need to prove the first line and prove going from the second line to the third line. The first line I was thinking would go like this. Let $\psi_n(\omega) = \int_S^T \phi_n(t,\omega)dB_t(\omega)$ and $\psi(\omega) = \int_S^T f(t,\omega)dB_t(\omega)$. Then by assumption $\psi_n\to\psi$ in $L^2(P)$. So $\mathbb{E}[(\psi_n - \psi)^2]\to 0$. Well $\mathbb{E}[(\psi_n^2 - \psi^2)^2] = \mathbb{E}[(\psi_n + \psi)^2(\psi_n - \psi)^2]$. At this point I have no idea how to proceed, mainly because I have no reason to think that $\psi_n^2, \psi^2\in L^2(P)$. The second to third line has a similar difficulty, but at least I know that the integral exists by the assumptions on $\mathcal{V}(S,T)$. This was just stated as a corollary as to the definition of the Ito integral for functions $f\in\mathcal{V}(S,T)$ in Oksendal with no direction given.