Expected value of global functions in renormalization group

This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $$C=C_{1}+\cdots+C_{N}$$ is a covariance matrix on $$\mathbb{R}^{|\Lambda|}$$. Besides, $$\varphi$$ is a random vector with distribution $$\mu$$ being Gaussian with covariance $$C$$; In addition, each $$\xi_{j}$$ is a random vector with distribution $$\mu_{j}$$ being Gaussian with covariance $$C_{j}$$, $$j=1,...,N$$. Now, on page 26, Brydges defines, for a given $$X\subset \Lambda$$, the set $$\mathcal{N}_{j}(X)$$ which is an algebra of functions measurable with respect to the $$\sigma$$-algebra generated by $$\{\varphi_{j}(x), x\in X\}$$. Here, $$\varphi_{j}(x) = \sum_{k>j}\xi_{k}$$ are random vectors. In my understanding, an element of $$\mathcal{N}_{j}(X)$$ is a real valued random variable defined on an underlying probability space, say $$(\Omega, \mathcal{F},P)$$. On page 27, Brydges defined $$F^{X}=\prod_{B\in \mathcal{B}_{j}(X)}F(B)$$. Again, $$F^{X}$$ seems to be a real valued function on $$(\Omega, \mathcal{F},P)$$. But equation (2.21) (Lemma 2.9) states that: $$\begin{eqnarray} \mathbb{E}_{j+1}F^{\Lambda} = (F')^{\Lambda} \end{eqnarray}$$ and $$\mathbb{E}_{j+1}$$ is defined on page 24 as an integral with respect to $$\mu_{j}$$ which is a measure on $$\mathbb{R}^{|\Lambda|}$$. How can $$F^{\Lambda}$$ be viewed as a function on $$\mathbb{R}^{|\Lambda|}$$?

I don't have time to go back and read exactly the definitions, but I suspect the issue here is that Brydges explicitly writes the dependence on the supporting sets, but not the dependence on the fields. I recommend that you review the definitions by adding/restoring in your notation this dependence in the fields, with care: namely don't blur the difference between $$\sum_{k>j}\xi_k$$ and $$\sum_{k\ge j}\xi_k$$ for example. I know it sounds silly, but if you do this carefully, you should see that some field remains which has not been integrated over. So the result $$(F')^{\Lambda}$$ should be a function of that remaining field. The latter could a priori be a function of blocks of some scale (coarse field), but you can see it as a function of unit blocks (fine field) in a trivial way: the value of the fine field for unit box is the value of the coarse field on the big block that contains the unit box.
• 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. – IamWill Feb 4 at 21:34