There is a choice to be made here: working with Banach spaces or with spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$. There are pros and cons for each of these two settings.
Let me stick to the original question and distributional setting. To learn the theory of probability measures on spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$, with a view to applications to constructive QFT, Glimm-Jaffe and Gel'fand-Vilenkin books are okay, but not that great. They miss the most fundamental theorem needed: the Lévy continuity theorem. Namely, it is the characterization of weak convergence of probability measures using the pointwise convergence of the generating function of Schwinger functions, e.g., when one removes a UV cutoff.
The closest to a convenient one-stop reference that I know is: "Generalized random fields and Lévy's continuity theorem on the space of tempered distributions" by Biermé, Durieu and Wang.