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 viwed as a function on $\mathbb{R}^{|\Lambda|}$?

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