I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-Landau model, Kupiainen introduces a spin configuration as a function $\phi: \mathbb{Z}^{d} \to \mathbb{R}$. Thus, we can view it as a sequence $\phi = (\phi_{x})_{x\in \mathbb{Z}^{d}} \in \mathbb{R}^{\mathbb{Z}^{d}}$. If $\Lambda \subset \mathbb{Z}^{d}$ is finite, we can consider the restriction $\phi_{\Lambda} = (\phi_{x})_{x \in \Lambda} \in \mathbb{R}^{\Lambda}$, where $\phi_{x}:= \phi(x)$. Because $\Lambda$ is finite, we can consider $\phi_{\Lambda}$ as an usual vector on $\mathbb{R}^{n}$ where $n$ is the cardinality of $\Lambda$. For instance, we can define the Gibbs measure on $\mathbb{R}^{\Lambda}$ as given by: \begin{eqnarray} d\mu_{\Lambda}(\phi) = \frac{1}{Z_{\Lambda}}e^{-H_{\Lambda}(\phi)}\prod_{j=1}^{n}d\phi_{x} \tag{1}\label{1} \end{eqnarray} where $Z_{\Lambda}$ is a normalizing factor, $H_{\Lambda}: \mathbb{R}^{\mathbb{Z}^{d}}\to \mathbb{R}$ is an Hamiltonian with some given boundary conditions and $\prod_{j=1}^{n}d\phi_{x}$ is just the product Borel measure on $\mathbb{R}^{\Lambda}$. On the other hand, on his new set of notes, on page 31 (also about Ginzburg-Landau model) Kupiainen states that "in classical statistical mechanics one considers $\phi(x)$ as a random variable with probability distribution given by (\ref{1})".Now, if I understood it correctly, this means that each $\phi_{x}$ is now a random variable on some underlying probability space. But then, the picture changes a lot, since now instead of simple vectors on $\mathbb{R}^{\Lambda}$, $\phi_{\Lambda}$ is a vector of functions. What does even mean to write $\prod_{j=1}^{n}d\phi_{x}$? Also, if this were the Ising model, we expect $\phi_{\Lambda}$ to be just a vector with entries $\pm 1$ as in the first picture. So, am I missing something here? Why sometimes $\phi_{\Lambda}$ are viewed as vectors and sometimes as vectors of functions? Also, what does $\prod_{j=1}^{n}d\phi_{x}$ mean if each $\phi_{x}$ is a random variable?