Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-algebra on $X$. My question is whether there exists a canonical choice of $(X,\mu,f)$ that depends only on $H$ and not "arbitrary" choices like choosing an orthonormal basis for $H$.

For an example of what I mean, here is a construction that only works when $H$ is finite dimensional. Let $X=H$, and for $h\in H$ consider the function $f_h\colon X\to\mathbb C$ given by $f_h(x)=\langle h,x\rangle$. Let $\mu$ be the unique Borel measure on $H$ satisfying $$\int_X e^{i f_h(x)}\ d\mu(x) =e^{-\|h\|^2/2}$$ for all $h\in H$, from which it follows that $h\mapsto f_h$ is an isometry. Note that the $\mu$-expectation of the function $\|x\|^2\colon X\to\mathbb C$ equals $\dim H$.

In fact when $\dim H=\infty$, the analogue of this construction yields a measure supported on "elements of $H$ with infinite magnitude", which can be made sense of rigorously as distributions - elements of the space $\Phi^*$ in a Gelfand triple $(\Phi,H,\Phi^*)$.

Thus, a close variant of my question is to ask whether there exists a canonical construction of a Gelfand triple using only the space $H$ as input.