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 Schwarz' 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.