Identifying a generic Hilbert space $H$ with an $L^2$ space on some measure space. This may be a stupid question, but I was wondering, if we are given an
infinite dimensional Hilbert space $H$, is it possible to find (or to
hypothesise that there's) a measure space $(M,\mathcal M,\mu)$ such
that $H$ coincides with $L^2(M,\mathcal M,\mu)$?
Apparently this is a basic resultm and the answer is yes.

Second question:
Let $H$ be a real Hilbert space, let  $W=\{W(h):h\in H\}$ be an isonormal Gaussian process, namely  a Gaussian Hilbert space indexed by $H$; then we know that $W=H^{:1:}$ (the first homogeneous chaos), and we further know that the stochastic Gaussian integral $I$ on a measure space $(M,\mathcal M,\mu)$  is an isometry from $L^2(M,\mathcal M,\mu)$ onto $H^{:1:}=W$.
Then any $W(h_i), h_i\in H$ can be represented as a stochastic Gaussian integral of some function in $L^2(M,\mathcal M,\mu)$, hence $W$ will be a Gaussian Hilbert space indexed by $L^2(M,\mathcal M,\mu)$.
My question is are $H$ and $L^2(M,\mathcal M,\mu)$ the same space?
If yes, the $W(\cdot)$ can be seen as a Gaussian stochatic integral.
Is it correct?
I may have made some mistakes on my reasoning, so if you spot something off please let me know!
Thanks in advance.
 A: You get an answer to your first question in the comments so let me address the second question. First I need to clean up some notation.
$H$ isn't itself a Gaussian Hilbert space so it seems incorrect to talk about the first homogeneous chaos over $H$. The situation is that the isonormal Gaussian process $W$ is an isometry $$H \to \tilde{W} = \tilde{W}^{\mathpunct: 1 \mathpunct:} = \operatorname{Ran}(W) \subseteq L^2(\Omega)$$ for some probability space $\Omega$ (Note I change some notation to avoid giving $W$ two definitions).
Now suppose we are given a linear isometry $I: L^2(M, \mathcal{M}, \mu) \to \tilde{W}$ (i.e. a Gaussian stochastic integral on $(M, \mathcal{M}, \mu)$ into $\tilde{W}$). Note that this is an assumption and is not automatically true. What you get for free is that there is a Gaussian Hilbert space $V$ such that there is a Gaussian stochastic integral $$\tilde{I}:L^2(M, \mathcal{M}, \mu) \to V.$$
In general, it is not possible to assume $V = \tilde{W}$. A simple way to see this is to notice that $\tilde{W}$ may be finite dimensional whilst $L^2(M, \mathcal{M},\mu)$ may be infinite dimensional so that there can be no such isometry. However, for the sake of discussion, I suppose we are given such a Gaussian stochastic integral into $\tilde{W}$.
The next thing to notice is that $I$ need not be surjective. I assume you are still following "Gaussian Hilbert Spaces" by Svante Johnson. It's worth noting that when introducing Gaussian stochastic integrals he says that we can "assume without essential loss" that $I$ is surjective. He does this by noting that the range of $I$ is a Gaussian Hilbert space and we can hence just restrict the codomain. In the situation of this question however we have fixed the Gaussian Hilbert space $\tilde{W}$ and so can't restrict the codomain in this way. This means that even assuming there is a Gaussian stochastic integral into $\tilde{W}$, you don't get that $I: L^2(M, \mathcal{M}, \mu) \to \tilde{W}$ is surjective automatically. As an example of this, we may have that $H$ (and hence $\tilde{W}$) are infinite dimensional but $L^2(M, \mathcal{M}, \mu) = \mathbb{R}$ (take $M = \{0\}$ and $\mu = \delta_0$) so that $I$ cannot be surjective and also $H \not \approx L^2(M,\mathcal{M},\mu)$.
For the sake of further discussion, I will now assume that $I: L^2(M, \mathcal{M}, \mu) \to \tilde{W}$ is additionally surjective. Now the situation is simple. $W: H \to \tilde{W} \subseteq L^2(\Omega)$ is an isometric isomorphism, as is $I: L^2(M, \mathcal{M},\mu) \to \tilde{W}$. Hence $I^{-1} \circ W: H \to L^2(M,\mathcal{M},\mu)$ is an isometric isomorphism. However, the assumptions we had to make to get here mean this is far from the generic situation.
