Stochastic PDE, infinite dimensional Hilbert Space [HARD]

I am studying the following stochastic PDE (from Damiano Brigo's book):

where is a stochastic process taking values in a Hilbert space H. G is the generator (of the strongly continuous semi-group of shifts) and is an infinite sequence of Brownian motions. One can take H to be the space of absolutely continuous functions from R_+ to R.

Here is my question: how do we address the question that each path maps the time-interval into an infinite dimensional Hilbert space ?! That is, the stochastic PDE generates an infinite dimensional path measure?

I think this is quite a challenging question.