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 $$ \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$. 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 $$ ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega) $$ and $$ ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$
Wikipedia claims the reference for the above is Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. However, I didn't find things like "$\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$" in the book.
My understanding of Itō isometry is that given fixed $T$, $\left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2}$ and $\int_{0}^{T} X_{t}^{2} \, \mathrm{d} t$ are both random variables not stochastic processes, and $\mathbb{E}$ denotes expectation with respect to the probability measure on the underlying probability space $\Omega$. Is my understanding correct?
Why does Wiki's Itō isometry treat $\left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2}$ and $\int_{0}^{T} X_{t}^{2} \, \mathrm{d} t$ as stochastic processes, and $\mathbb{E}$ as expectation with respect to classical Wiener measure $\gamma$ which is a measure on the functional space $\mathbb{R}^{[0,T]}$ induced by the Wiener process $W$? I am not sure if Wiki is consistent with itself, because in the last two formulas for the inner products, the integrals are wrt the classical Wiener measure over the underlying probability space $\Omega$ instead of over the functional space. Or do I misunderstand Wiki?
Is Wiki's Itō isometry a different version from the the one that I understand? If yes, is there some reference for Wiki's version and relation between the two?
Thanks and regards!
Since W is the canonical Wiener process you have right from the definition that $\Omega$ is equal to the space of functions from the index set to the state space (usually denoted by $C_0$), i.e.
$$(\Omega,\mathcal{A},\mathbb{P})=(C_0,\mathcal{B}(C_0),\mu)$$
where $\mu$ denotes the Wiener measure. So Wiki is consistent with itself.
Moreover, Wiki does not treat $\left( \int_0^T X_t \, dW_t \right)^2$ and $\int_0^T X_t^2 \, dt$ as stochastic processes, but as random variables
$$C_0 \ni \omega \mapsto \left( \int_0^T X_t \, dW_t \right)^2(\omega) $$
resp.
$$C_0 \ni \omega \mapsto \left(\int_0^T X_t^2 \, dt \right)(\omega) $$