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I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. I would like to learn more about these measures so I can read some of the CQFT literature.

There is an older post (Measure theory in nuclear spaces) that asks for a similar reference and I have gone through most of the references suggested there. I have read Gelfand's book, and though the exposition is great, I find it presents the material in a way that is no longer standard today (please correct me if I am wrong here). I have also read the section suggested in B. Simon's book, but it was not much more than a very quick introduction.

As part of my search I have found a number of references that go over Gaussian measures in great depth, such as books by Bogachev or Strook. However these books go into much more detail than I need.

To summarize the above, I am looking for a good reference that goes into as much detail as necessary so that one can begin reading the CQFT literature right after. If this does not exist, can someone suggest what topics are needed in Gaussian measure theory for my goal?

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    $\begingroup$ In every single case that I am aware of, these measures can be realised on some negative Sobolev space (possibly weighted) so you really just need measures on Hilbert (or at most Banach) spaces. Chapter 4 of my SPDE notes already contains more than what you'll need. $\endgroup$
    – Martin Hairer
    Commented Feb 7 at 7:10
  • $\begingroup$ @MartinHairer Thank you, I'm very excited to go through these notes! $\endgroup$
    – CBBAM
    Commented Feb 7 at 7:27

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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

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  • $\begingroup$ Thank you very much for the resources! $\endgroup$
    – CBBAM
    Commented Feb 23 at 2:34

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