A good reference, for practical purposes, is Section I.2 of the book "Functional Integration and Quantum Physics" by Barry Simon.


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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 <a href="http://www.numdam.org/item/AIF_1967__17_1_1_0">"Processus lin&eacute;aires, processus g&eacute;n&eacute;ralis&eacute;s"</a> by Fernique. I have also seen references to <a href="">a book by Dalecky and Fomin</a>, but I don't have access to a copy.