# Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far from being normed as possible" [nLab], and I haven't been able to find a reference in this setting.

Is there a good reference on measure theory in nuclear spaces?

• I'm sure you know about Bogachev's Gaussian Measures, much of which works in the context of possibly nuclear spaces. Bogachev writes extensive bibliographies, so perhaps you will find something there which discusses measure theory in linear spaces more generally. – Nate Eldredge Nov 15 '13 at 18:42
• Thanks, Nate! Bogachev's books are great, so it's a good suggestion to check the references there. – Tom LaGatta Nov 15 '13 at 19:31
• Perhaps more informative than "nuclear spaces are as far as possible from normed spaces" is the factoid combo: 1) infinite dimensional normed spaces are as far as possible from finite dimensional spaces, and 2) infinite dimensional nuclear spaces are as close as possible to finite dimensional spaces. – Abdelmalek Abdesselam Dec 30 '17 at 23:31

## 3 Answers

This subject was studied in some detail in the 70's by the French school, in particular in light of Minlos' theorem and cylindrical measures. I remember it being a favourite topic in the celebrated "Seminaires". You can find material in "Radon measures on topological spaces and cylindrical measures" by L. Schwartz, presumably with more detailed references.

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.

The fourth volume of I. M. Gel'fand's "Generalized Functions", subtitled "Applications of Harmonic Analysis" (written together with N. Ya. Vilenkin and published by Academic Press in 1964, now recently reissued by AMS-Chelsea) discusses this topic abstractly and in depth. It is, in fact, the standard reference for the subject as cited in other books such as J. Glimm's and A. Jaffe's "Quantum Physics - A Functional Integral Approach" (2nd. ed., Springer-Verlag, 1988) and L. Schwartz's book cited in alpha's answer.

It must be said, though, that cylinder set measures as they appear e.g. in the important Bochner-Minlos theorem are actually built over topological duals of nuclear locally convex vector spaces (lcvs).

• One issue with the Gelfand approach is that it only talks about countably Hilbert nuclear spaces. Grothendieck's definition is more general and probability theory should work well in this more general setting too. – Abdelmalek Abdesselam Dec 30 '17 at 21:19
• Gel'fand's approach actually considers more general (topological duals of nuclear) spaces too, but you may loose countable additivity for cylinder set measures. This is true for Schwartz's approach as well. However, you do get countable additivity just like in the case of nuclear countably Hilbert spaces if the (underlying TVS of the) measure space is the topological dual of a countable inductive limit of countably normed nuclear spaces - which includes $\mathscr{D}'$. In fact, Section IV.2 of Gel'fand-Vilenkin discusses all relevant cases for countable additivity of cylinder set measures. – Pedro Lauridsen Ribeiro Dec 30 '17 at 22:48
• I agree with the letter of what you said. My contention is with the spirit of the presentation by Gelfand et al which I find wrong (it's a personal opinion). I think if you loose countable additivity the measure belongs in the garbage can. Also, cylinder set measures are the wrong concept. We should be talking about Borel probability measures. Glimm and Jaffe, perhaps influenced by Hida or Gelfand et al, made a similar methodological error. – Abdelmalek Abdesselam Dec 30 '17 at 23:28