Definition of $L_2(\Omega \times [0,T])$ - Stochastic Processes-

Question

Can somebody give me a precise and clear definition of this space: $L_2(\Omega \times [0,T])$?
I failed to find one on the net.
Is this the space where: $\forall t=t_0 \;fixed \; \in [0,T]$ we have:
$E(X^2(t_0,\omega))=\int_{\omega \in \Omega}^{}X^2(t_0,\omega)d\mathbb{P}(\omega)<\infty $
And $\forall \omega=\omega_0 \;fixed \; \in \Omega$ we have:
$\int_{0}^{T}X^2(t,\omega_0)dt<\infty $

Answer 1

$L_2(\Omega \times [0,T])$ is the space of equivalence classes of functions $X(\omega,t)$ that satisfy

$$ \int_0^T \int_{\omega \in \Omega}^{}X^2(\omega,t)\, d\mathbb{P}(\omega)\,dt <\infty \,.$$

The equivalence relation is equality a.e. with respect to the product measure $\mathbb{P} \times \text{Lebesgue}_{[0,T]}.$

The order of integration does not matter by Tonelli's theorem. the inner product on this space is, as usual in $L^2$ spaces, $$\langle X,Y \rangle :=\int_0^T \int_{\omega \in \Omega}^{}X (\omega,t)Y(\omega,t) \, d\mathbb{P}(\omega)\,dt \,.$$ (If the functions are allowed to take complex values then $Y$ should be conjugated.)