It was so obvious! Thank you so much! It all makes sense now!

Thanks for your answer! I followed your advice but I'm still a little confused. As you said, $(F')^{\Lambda}$ should be a function of the remaining fields but this is actually a composite function, as it seems. For instance, the function $F^{\Lambda}$ should be defined over $\Omega$ but it can be written as $F^{\Lambda} = f(\zeta_{j+1},\psi_{j+1})$ for some function $f:\mathbb{R}^{2\Lambda}\to \mathbb{R}$, so that $F^{\Lambda}(\omega) = f(\zeta_{j+1}(\omega),\psi_{j+1}(\omega))$. But Brydges integrates $F^{\Lambda$ over $\mathbb{R}^{\Lambda}$ as some sort of functional integral. I'm lost here.

Oh, good to know! I thought I was missing something!

Is this just abuse of notation or both things are essentially the same?

It's getting clearer to me. There is just one thing that is bothering me: he seems to be using variables $\varphi$ interchangeably as a vector in $\mathbb{R}^{|\Lambda|}$ and as a random vector. For example, in page 10 he writes that, for a positive-definite matrix $A$ in $\mathbb{R}^{|\Lambda|}$, $\mu := \mbox{const} e^{-\frac{1}{2}\langle \varphi, A^{-1}\varphi\rangle}$ is a Gaussian measure. Thus $\varphi \in \mathbb{R}^{|\Lambda|}$ as I understand. Then, he writes $\varphi \sim N(C)$, where $C=A^{-1}$ and here $\varphi$ is a random vector $\varphi: \Omega \to \mathbb{R}^{\Lambda}$.

Thanks for your answer! Now I can spot my misunderstanding. But then, for each $j $, $\varphi_{j}(x) $ is a function, right? What is that function? I don't seem to understand how this emerge from $\varphi_{j}=\sum_{k>j} \xi_{k}$.

Actually is the Brydges lecture notes on RG. I will add it to the post.