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 cand be found in *Laurent Schwarz*' book *Radon measures*.

Thinks are pretty  reasonable for Gaussian measures  defined on duals of nuclear spaces.  The space of distributions (generalized functions)  on aan 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  covaraince operatro $\mathscr{K}$.  

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