I mean, once you worked out the details and defined an infinite volume measure from the discretized version of the model, you get that the measure on $\mathcal{S}'(\mathbb{R}^{d})$ defined by $(-\Delta+m^{2})$ is the 'appropriate' infinite volume measure, but then these measures can only be 'interpreted' a posteriori.

(...) once you induced these infinite volume measures from the discretized model. Is it true, indeed?

Just a quick question: as I said, I will work the details later but as far as I understood it by reading it quickly, you proceed by taking limits to construct the infinite volume measures. This is something that confuses me a lot: if I want to work directly on $\mathbb{R}^{d}$. I know I can define (by Minlos-Bochner) the Gaussian measure on $\mathcal{S}'(\mathbb{R}^{d})$ with covariance $(-\Delta+m^{2})$, but it is not clear to me that this has to do with my "measure" (\ref{1}) in the first place. My point is: as far as I know, these connections are only possible once you induced (cont)

Thanks again for the answer! I'm in a hurry, so I can't work out the details right now, but as usual this is an exellent/very detailed answer so I already accepted it because the bounty was going to expire in a few hours.

@CarloBeenakker thanks for the reference! What do you mean by 'derive this from some physical model'? You mean choosing the right covariance so it represents some physical model?

@AbdelmalekAbdesselam I really enjoyed your answer on the question linked. You said a general theory might be too much to ask, and I already expected that. But what about the abstract formulation I cited? Sounds pretty general to me.

@IgorKhavkine you are absolutely right! I will make and edit! Thank you!

Nice! I'm going to take a look at that! Thanks!!

BTW, I also did a very extensive search on the internet about materials discussing these sequence spaces $s$ and $s'$ but I didn't find anything really helpful. I think this is because of the identifications of $s$ and $s'$ to $\mathcal{S}$ and $\mathcal{S}'$, and people prefer to work with the latter. Also, I don't think these sequence spaces have a proper name on the literature, so it is difficult to search. Just out of curiosity, do you know any material on these sequences?

@AbdelmalekAbdesselam thanks again for the answer! Indeed, the problem is related to these spaces of sequences you mentioned. I'm working out the details of your previous answers and, in special, the realization of $s'$ as the dual of $s$. I realized that when you said the dual was equiped with the strong topology, I automatically assumed that this was the strong operator topology on Simon & Reed's book, but the underlying spaces there are both Banach. I tried to look for equivalences on the internet but I didn't find anything closely related, only alternative constructions.