That's a very nice comment. The point is: how to systematically define what we mean by a infinite volume limit, in order to study the system in the thermodynamic limit? The Gaussian measure $\mu$ on $\mathbb{R}^{\mathbb{Z}^{d}}$ constructed using Kolmogorov's Theorem is a legit infinite volume limit but, in general, the infinite volume measure is assumed to be that satisfying (2). It then seems that the limit is taken conveniently, depending on the model rather than a systematic approach.

Edited! Thanks!!

Thanks for your comments and references. Let me ask you something: in general, the sine-gordon is used as I stated in only formal way? I hardly never see different approaches.

Of Course, there are some intersections between these two questions. The answer I got in the prévios question was great, but the authors seem to understand these topics in a different way (none of them, for instance, make explicit use of formal series) so I'm trying to understand what am I missing and how should I read all these properly.

Yes. Let me elaborate. In the first question I tried to understand the mathematical meaning of (2). This is because I thought there might be such meaning. At the moment, I think the Sine-Gordon transform is meant to be formal, as many papers state. But this raises other questions such as (1) what's the point of such a formal definiton, (2) is this the only construction?

Can you elaborate a little more, when you have a chance? It seems that this "space of continuous functions" must be, in fact, a schwartz space. I didn't follow you when you said to convolve $\psi$ with some function to get $\phi$.

Thanks for your answer! Just to clarify: I set $\Omega = \cup_{N}\Lambda^{N}$ to be my configuration space and I can take the Borel $\sigma$-algebra on each $\Lambda^{N} \subset \mathbb{R}^{dN}$, right? With what $\sigma$-algebra should I equip $\Omega$?

Ok, I think I get the point. It is just a regular series of real numbers. I thought it would be more natural to define it in terms of some sort of weak convergence of some measure in a "bigger space". Thank you!!

In some references, the $\sigma$-algebra is assumed to be $\cup_{N}\Gamma_{N}(\Lambda)$ where $\Gamma_{N}(\Lambda) := \{(x_{1},...,x_{N})\in (\mathbb{R}^{d})^{\Lambda}, x_{i}\in \Lambda\}$ but I don't know if this is the case.

@AbdelmalekAbdesselam this time I'm not following Brydges notes. But I would expect $\mu(x_{1})\times \cdots \times \mu(x_{N})$ to be a measure on, I don't know, $\mathbb{R}^{\infty}$ and $\int_{\Lambda}$ to be just the integral restriced to $\Lambda$. Adapting the answer would lead to a $\sigma$-algebra consisting on $2^{\Lambda}$, that is, every subset of $\Lambda$ is measurable, right?

The case $\Lambda$ finite is assumed when we're dealing with $\mathbb{Z}^{d}$ rather than $\mathbb{R}^{d}$. Your answer seems to address the $\mathbb{Z}^{d}$ case and it is still very useful to me because I get in troble understanding the problem in $\mathbb{Z}^{d}$ a well. But my original post should have considered $\Lambda$ bounded and I didn't notice my mistake.