This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations:

**(1)** $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, respectivelly, the single particle configuration space and its associated $\sigma$-algebra.

**(2)** If $\Lambda \subset \mathbb{Z}^{d}$ is finite, $\Omega_{\Lambda} := \{-1,1\}^{\Lambda}$ and $\mathcal{F}_{\Lambda} := \bigotimes_{x\in \Lambda}\mathcal{F}_{0}$ is its associated product $\sigma$-algebra.

**(3)** $\Omega := \{-1,1\}^{\mathbb{Z}^{d}}$, $\mathcal{F} := \bigotimes_{x\in \mathbb{Z}^{d}}\mathcal{F}_{0}$.

In what follows, I'm following Friedli and Velenik's book, chapter 3. For finite-volume systems, we can define Gibbs distributions with free boundary conditions (according to Definition 3.1 of the mentioned reference) by: \begin{eqnarray} \mu_{\Lambda,\beta,h}^{\emptyset}(\{\omega\}):= \frac{1}{Z_{\Lambda,\beta, h}^{\emptyset}}e^{-\beta H_{\Lambda,\beta,h}^{\emptyset}(\omega)} \tag{1}\label{1} \end{eqnarray} this is a discrete measure on $\Omega_{\Lambda}$. Also, if we fix $\eta \in \Omega$, we can define (according to Definition 3.3) Gibbs states with $\eta$-boundary conditions: \begin{eqnarray} \mu_{\Lambda,\beta,h}^{\eta}(\{\omega\}):= \frac{1}{Z_{\Lambda,\beta, h}^{\eta}}e^{-\beta H_{\Lambda,\beta,h}^{\eta}(\omega)} \tag{2}\label{2} \end{eqnarray} This, on the other hand, is a discrete measure on $\Omega_{\Lambda}^{\eta}:= \{\omega \in \Omega: \hspace{0.1cm} \mbox{$\omega_{x} = \eta_{x}$ for all $x \in \Lambda^{c}$}\}$ (which is equipped with its discrete $\sigma$-algebra).

The main ideia of the theory is to study thermodynamic limits, both for thermodynamic quantities and Gibbs states. In the case of Gibbs states, thermodynamic limits means weak convergence of finite-volume Gibbs states (at least in the present context, where $\Omega_{0}=\{-1,1\}$. This is, in fact, the main reason for which $\mu_{\Lambda,\beta,h}^{\eta}$ is conveniently defined on $\Omega_{\Lambda}^{\eta}$ instead of $\Omega_{\Lambda}$. Once we'd like to study Gibbs states on $\Omega$ using weak-convergence of finite volume Gibbs states, we need to extend our finite volume Gibbs states to equivalent notions that live in 'the whole space' $\Omega$. It is easy to do that with the measure (\ref{2}), since we can define (with abuse of notation): \begin{eqnarray} \mu_{\Lambda, \beta,h}^{\eta}(\{\omega\}) = \begin{cases} \displaystyle \frac{1}{Z_{\Lambda,\beta, h}^{\eta}}e^{-\beta H_{\Lambda, h}^{\eta}(\Pi_{\Lambda,\eta}\omega)} \quad \mbox{if $\omega_{x} = \eta_{x}$ for all $x\in \Lambda^{c}$} \\ \displaystyle 0 \quad \mbox{otherwise} \end{cases} \tag{3}\label{3} \end{eqnarray} where $\Pi_{\Lambda,\eta}$ is the canonical projection $\Omega \hookrightarrow \Omega_{\Lambda}^{\eta}$. Note that this simple procedure does not work for extending $\mu_{\Lambda,\beta,h}^{\emptyset}$, since this would require to define $\eta$ as zero outside $\Lambda$, which is not consistent with our definition of $\Omega_{0}$.

Now, in this context, R. Ellis defines (I think this is standard, tho) $\mathcal{G}_{0}(\beta,h)$ as the set of all weak-limits of measures $\mu_{\Lambda_{n},\beta,h}^{\eta_{n}}$, where $(\forall n)$ $\eta_{n}\in \Omega$ and $\Lambda_{n}\to \mathbb{Z}^{d}$ is an increasing sequence of finite subsets of $\mathbb{Z}^{d}$. Furthermore, we define: \begin{eqnarray} \mathcal{G}(\beta,h) := \overline{\mbox{conv}\mathcal{G}_{0}(\beta,h)} \tag{4}\label{4} \end{eqnarray} where $\mbox{conv}\mathcal{G}_{0}(\beta,h)$ stands for the convex hull of $\mathcal{G}_{0}(\beta,h)$.

**Question(s):** First, I'd like to know how to extend $\mu_{\Lambda,\beta,h}^{\emptyset}$ to 'the whole space' $\Omega$ as done in (\ref{3}), since we cannot take zero values on $\Omega_{0}$. Also, I'd like to understand the role of these measures $\mu_{\Lambda,\beta,h}^{\emptyset}$ on $\Omega$: there are some results on weak-convergence of these measures in the theory, which seems to indicate they play some important role after all, but they seem **not being considered** in the definition of $\mathcal{G}_{0}(\beta,h)$, since we're only taking $\eta_{n}\in \Omega$ as boundary conditions of the sequence of Gibbs states $\mu_{\Lambda_{n},\beta,h}^{\eta_{n}}$. So, what am I missing here? Can we go on and study all (or at least almost all) relevant issues of the theory without considering infinite volume measures $\mu_{\Lambda,\beta,h}^{\emptyset}$? I know these measures are relevant to study, say, correlation inequalities and so on, but these do not demand them to be defined in the whole $\Omega$.