Construct identification between $L^2( \Omega;H)$ and $L^2(T \times \Omega)$ where $H=L^2(T, \mathcal B,\mu)$

Question

I am reading page 31 of Nualart, "The Malliavin Calculus and Related Topics" .

Here, it says that there is an identification between the Hilberts spaces $L^2( \Omega;H) $ and $L^2( T \times\Omega) $ where $H=L^2(T, \mathcal B,\mu)$ and $\mu$ is atomless.

How can I contruct this "natural identification"? I know that $L^2(T \times \Omega) \cong L^2(T) \bigotimes L^2(\Omega)$, but I don't see how this helps.

Any hint is appreciated.

Answer 1

The map is just $f \in L^{2}(\Omega, H) \to g$ where $g(t,\omega)=f(\omega) (t)$. The fact that this is an isometry follows immdiately from Fubini's Theorem.