A good reference, for practical purposes, is Section I.2 of the book "Functional Integration and Quantum Physics" by Barry Simon.
Edit: the above is good for a very light introduction. But in order to go further into probability theory on spaces like $\mathcal{S}, \mathcal{S}',\mathcal{D},\mathcal{D}', \oplus_{\mathbb{N}}\mathbb{R}, \prod_{\mathbb{N}}\mathbb{R}$, etc. I think the best reference is the article "Processus linéaires, processus généralisés" by Fernique. I have also seen references to a book by Dalecky and Fomin, but I don't have access to a copy.