Itō isometry from Wikipedia:
Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener process. Then $$ \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right], $$
In other words, the Itō stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products $$ ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \mathrm{d} \mathrm P (\omega) $$ and $$ ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \mathrm P (\omega). $$
Since it suggests an isometry between two normed spaces, I was wondering
- On the LHS, why is the ito integral $\int_{0}^{T} X_{t} \, \mathrm{d} W_{t} $ a $L^2(\Omega)$ function, so that we can talk about its $L^2$ norm?
- On the RHS, does $\mathrm E (\int_0^T X_t^2 dt)$ equal $\int_{\Omega \times [0,T]} X^2 d(\mathrm{P} \times \lambda)$, and $X \in L^2(\Omega \times [0,T])$, so that we can talk about the $L^2(\Omega \times [0,T])$ norm of $X$? Here $\mathrm{P}$ is the probability measure on $\Omega$, $\lambda$ is the restrction of the Lebesgue measure on $[0,T]$, and $\mathrm{P} \times \lambda$ means their product measure. I can't apply Fubini't theorem here, because its conditions seem not apply here.
Note that in Wikipedia version of Ito isometry, the process $X$ is only required to be adapted to the filtration of the Wiener process. In Shreve's Stochastic Calculus for Finance, the Ito isometry is under the asssumption that $\mathbb{E} [\int_0^T |X(t)|^2 dt] < \infty$. Under both cases, I am not able to figure out the answers to my questions above.
Thanks and regards!