In the book Kazhdan’s Property (T) (Appendix A7) by Bekka, de la Harpe and Valette the symmetric Fock space on a Hilbert space is $H$ studied as the analogue of a space of measurable functions on a Hilbert space $H$. This is called the Gaussian construction and quite important if one wants to pass from unitary representations of a group $G$ to actions of $G$ on a probability measure space. This is probably not quite what you want, but serves as a suitable replacement of the the Gaussian measure (on a finite-dimensional Hilbert space) for many purposes.
In case $H$ is finite-dimensional, it precisely corresponds to the study of the Gaussian measure on $H$. Here, the correspondence is clear: If $G$ acts by unitary operators on $H$, then it preserves the Gaussian measure $\mu$ on $H$ and hence, there is an associated action on the probability space $(H,\mu)$.