I have already addressed this problem on my previous question but I still have trouble understanding Brydges' RG maps on his lecture notes, 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}).


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 $\mathbb{R}^{|\Lambda|}$.

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  • $\begingroup$ It was so obvious! Thank you so much! It all makes sense now! $\endgroup$ – IamWill Feb 6 at 11:11

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