Awesome answer! Thanks!!

Thak makes sense. But is it just a discretization or one also decomposes fields?

@gmvh I'm a student in statistical mechanics but know hardly anything about QFT. I knew that block spins transformations were developed for statistical mechanics models by Kadanoff and, then, Wilson, but I didn't know if QFT used it for some reason. Thanks for the comment!

ahh! Sorry I misunderstood it. Now I got it!

Thanks for the answer! Just to clarify, when you say (\ref{2}) is perfectly well defined for $\varphi $ in the subset of $\mathcal{S}(\mathbb{R}^{d})$ you mean $\mathcal{S}'(\mathbb{R}^{d})$ via the injection map you defined?

apart from that, I've seen some books where even thins like $\int_{\mathcal{S}'(\mathbb{R}^{d})}\varphi(f)^{n}d\mu_{G}(\varphi)$ were written as $\int \varphi(x)^{n}d\mu_{G}(\varphi)$, so the pointwise notation was used again but just as a matter of notation for something that has meaning. This led me to believe that the $\varphi(x)^{4}$ was also a notation convention or something, as I said, to idk see field variables as random variables or something. But from the content of the comments it seems that this is a formal convention which we cannot avoid, maybe because of the physics of it (?).

(cont) the addition of a term such as $\int_{\mathbb{R}^{d}}\varphi(x)^{4}dx$ which shall be treated as a perturbation of the former Gaussian measure is, in this case, quite strange to me since you worked hard to give precise meaning to the Gaussian measure etc and then you add some formal object to the theory. I mean, to my understanding, this has two possible explanations: (a) this is motivated by the physics behind it and there's not much we can do or (b) this is just a matter of notation

As far as I understand (and I might be wrong) if you take $g=0$ in the action (\ref{2}), you can derive a rigorous (Gaussian) measure $\mu_{G}$ on $\mathcal{S}'(\mathbb{R}^{d}$ whose covariance is $-\Delta+m^{2}$, so this is a Gaussian measure associated to the action $S$. Of course, this has a problem which is that moments such $\int_{\mathcal{S}'(\mathbb{R}^{d})}\varphi(f)^{n}d\mu_{G}(\varphi) = \infty$, so we need to regularize the theory. (continues)

I'm happy my question gathered such a nice set of comments. I'm learning a lot already. Let me say some words above to adress you all at once.

@AaronBergman my point here is basically the pointwise dependence of tempered distributions. It is a very common language (I've encountered it in many notes, papers etc) in QFT and I don't know the real benefit of it, since it is clear formal. Also, you're correct, I'm seeing it as a perturbative theory. About the lattice regularization, I don't know how rigorous those continuum limits can be (as I said, I'm not an expert) but the formulation on the continuum causes me trouble, as I pointed out, when tempered distributions are represented as 'real valued functions'.

@DmitriPavlov you are right! I didn't even noticed it. Thanks. Well, this reinforces the statements in my post, right?