You ought to have a look at the $4$-th volume of *Gelfand-Vilenkin* on *Generalized Functions* where they describe this concept  in great detail, albeit in an old-fashion language.  The most comprehensive description I know can be found in *Laurent Schwartz*' book *Radon measures*.

Things are pretty  reasonable for Gaussian measures  defined on duals of nuclear spaces.  The space of distributions (generalized functions)  on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined  on a space of generalized   functions, but it is supported on a much "thinner" space. 


Beyond duals of nuclear spaces you need to  assume   some things about the  covariance operator $\mathscr{K}$.  

In any case,  have a look  at the above two references. 

**Edit:** The book **Gaussian measures** by Bogachev is also a very  good source.

**2023 Edit** There is a new (2023) book by Stroock **Gaussian measures in Finite and Infinite Dimensions** that does a particularly good job of explaining Gaussian measures and   some and their uses.  It  is not as general as Bogachev's book (only deals with Banach spaces) but it uncovers a  lot of the "mystery" of this concept.