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.

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, the Glimm-Jaffe and Gel'fand-Vilenkin books are okay, but not that great. They miss the most fundamental theorem needed: the Lévy Continuity Theorem (LCT). 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.

If one has the LCT, one does not even need the Bochner-Minlos Theorem, in order to construct the free field Gaussian measure. This was explained in my previous MO answers

Reformulation - Construction of thermodynamic limit for GFF

and

https://mathoverflow.net/a/365144/7410

A brief review of the precise statement of LCT and related results is in this previous post

https://mathoverflow.net/a/384168/7410

There is an important fact about probability measures on spaces of distributions which I don't think is covered in the article by Biermé et al., nor the GJ and GV books: if moments exist, they are automatically jointly continuous, see:

https://physics.stackexchange.com/a/802464/7266