We consider the isonormal Gaussian process $W=\{W(h),h\in H\}$ indexed by a separable Hilbert space $H$, defined on a complete probability space $(\Omega, \mathcal F,P)$ where $\mathcal F:=\sigma(W)$, i.e. $W(h)$ is a centered Gaussian random variable and the covariance structure is defined by $$E(W(h)W(g))=\langle h,g\rangle_H.$$
For example $W$ is the Wiener-Ito integral, $H$ is (a deterministic) $L^2([0,T])$ space and $(\Omega,\mathcal F,P)$ is the Wiener probability space.
Another example could be for instance $(\Omega,\mathcal F,P)=(\mathbb R,\mathcal B(\mathbb R),\nu)$ where $\nu$ is the standard Gaussian measure, $H=\mathbb R$, and $W(h)(x)=h\cdot x, \forall h\in H$.
In this two examples we can see that the probability space and the associated Hilbert space are of the "same type"; I mean, in the first example $\Omega$ and $H$ and function spaces, in the second one $\Omega$ and $H$ are one dimensional spaces (or finite dimensional in general).
I've been wondering for quite some time (and it may perfectly be a non-sense question) if for instance we can have $H$ to be an $L^2$ space and $\Omega$ a finite dimensional space?
More generally, is there some relation that we must satisfy between the space $\Omega$ and the space $H$? (being both function spaces or finite dimensional, etc)
Thanks in advance and I apologice if this is something trivial.