I have already addressed this problem on my [previous question](https://mathoverflow.net/questions/351921/expected-value-of-global-functions-in-renormalization-group) but I still have trouble understanding Brydges' RG maps on his [lecture notes](http://www.math.ubc.ca/~db5d/Seminars/PCMILectures/lectures.pdf), so I'll try to elaborate my question a little better.


Let $\Lambda \subset \mathbb{Z}^{d}$ be finite. We have a positive-definite $|\Lambda|\times |\Lambda|$ matrix $C$ which can be decomposed as a sum $C=C\_{1}+\cdots+C\_{N}$ where, for each $j=1,...,N$, $C\_{j}$ is again positive-definite. Let $\varphi:\Omega \to \mathbb{R}^{|\Lambda|}$ be a random vector $\varphi = (\varphi(x))\_{x\in \Lambda}$ with distribution $\mu\_{C}$ being a Gaussian measure on $\mathbb{R}^{|\Lambda|}$ associated to $C$. Moreover, for each $j=1,...,N$, $\xi\_{j}:\Omega \to \mathbb{R}^{|\Lambda|}$ is a random vector $\xi = (\xi(x))\_{x\in \Lambda}$ whose distribution $\mu\_{j}$ is a Gaussian measure associated to $C\_{j}$. In addition, to each $j=0,1,...,N$ we define new random vectors as partial sums $\varphi\_{j} :=\sum\_{k>j}^{N}\xi\_{k}$, where $\varphi\_{N}:=0$ and $\varphi\_{0}:=\varphi$. 


Given $X\subset \Lambda$, let $\mathcal{N}\_{j}(X)$ be the algebra of functions measurable with respect to the $\sigma$-algebra generated by $\{\varphi\_{j}(x), x\in X\}$ and $\tilde{\mathcal{N}}\_{j}(X)$ is the algebra of functions measurable with respect to the $\sigma$-algebra generated by $\{\xi\_{j+1}(x),\varphi\_{j+1}(x), x\in X\}$. If $X=\Lambda$, we write $\mathcal{N}\_{j}(\Lambda) \equiv \mathcal{N}\_{j}$ and similarly for $\tilde{\mathcal{N}}\_{j}$. 


Now, take $F^{\Lambda}$ to be a function which is $\tilde{\mathcal{N}}\_{j}$-measurable (equation (2.20) in Brydges notes). Then, Brydges evaluates the expectation:
\begin{eqnarray}
\mathbb{E}\_{j+1}F^{\Lambda} \tag{1}\label{1}
\end{eqnarray}
which should be just the expectation with respect to the measure $\mu\_{j+1}$. This is where I got in trouble: the measure $\mu\_{j+1}$ is on $\mathbb{R}^{|\Lambda|}$, but as far as I'm concerned, $F^{\Lambda}$ is a random variable defined on some underlying probability space, whose $\sigma$-algebra is generated by $\{\xi\_{j+1}(x),\varphi\_{j+1}(x),x\in\Lambda\}$. So I don't understand the meaning of (\ref{1}).


My guess is that Brydges is using some sort f isomorphic version of the actual $F^{\Lambda}$. In fact, because $F^{\Lambda}$ is $\tilde{\mathcal{N}}\_{j}$-measurable, there exists a function $f: \mathbb{R}^{2|\Lambda|}\to \mathbb{R}$ such that $F^{\Lambda}(\cdot) = f(\xi\_{j+1},\varphi\_{j+1})(\cdot)$ and this $f$ can be integrated with respect to $\mu\_{j+1}$ as in (\ref{1}). But I don't know if my guess is correct and, if it is, I don't know how to justify why can $F^{\Lambda}$ be "replaced" by $f$ in (\ref{1}). 



Renormalization group map on hierarchical models
I think Brydges is (tacitly) assuming that $\Omega=\mathbb{R}^{|\Lambda|}$. This, of course, is bound to create confusion between elements of $\mathbb{R}^{|\Lambda|}$ and random elements of $\mathbb{R}^{|\Lambda|}$. 


