@IgorKhavkine in summary, $\bf{\Psi}$ would be a random vector on $\mathbb{R}^{\Lambda}$ whose distribution is given by (\ref{1})? He just used the same notation for the random vector and vectors on $\mathbb{R}^{\Lambda}$.

@MichaelEngelhardt as far as I understand, saying that $\phi(x) = \phi_{x}$ is a random variable means that $\phi_{\Lambda} = (\phi_{x})_{x\in \Lambda}$ is a family of random variables indexed by $x \in \Lambda$, and $\phi_{x}: \Omega \to \mathbb{R}$ is a measurable function on some underlying probability space. So each entry of $\phi_{\Lambda}$ is a measurable function.

@DanieleTampieri thank you for the answer! You mentioned the answer to my previous question on the functional derivative. In this case (where $K\in \mathcal{S}'(\mathbb{R}^{d})$ is a derivative of some function $f: \mathcal{S}(\mathbb{R}^{d}) \to \mathbb{C}$, can we garantee the integral representation? This was my first post and it seems we can, but i not sure yet.

This is precisely what concerned me: a lot of authors use the integral representation and this'd confuse me. You explanation is perfect and it was my first guess. Thank you!

Great! Thank you so much for the help!

Thanks for the answer. Your first statement says that a continuous multilinear map of $\mathcal{S}(\mathbb{R}^{d}) \to \mathbb{C}$ is given by a tempered distribution on $\mathbb{R}^{nd}$. I've never found this result in books (only with $n=2$). Does it follow by induction on the case $n=2$?

I edited it. I think it's better now. Thanks!

Yes, you're correct. I think I should clarify this in the post, indeed. My relation (2) is just to stress that $f[\varphi]$ is continuous and linear for each $\varphi$ but my question is whether, for a fixed $\varphi$, there is such kernel $K_{\varphi}$. I'm really not interested in the dependence of $\varphi$.

@paulgarrett $f[\varphi]$ is multilinear. As an example, take $f[\varphi]$ do be the $n$-th derivative of $f$ at $\varphi \in \mathcal{S}(\mathbb{R}^{d})$.

@DanieleTampieri thanks so much!! Seems an amazing reference!

As a final comment, I'd like to put Glockner and Neeb's Theorem in terms of my definition of Gâteaux differentiability. I think these definitions are equivalent. Do you agree?