Spins in classical statistical mechanics

Question

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?