I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$. Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator $$ \mathcal{L}_{A}(\psi)(x)=\int_{\mathbb{S}^1} e^{A(ax)} \psi(ax)\ da $$ where $x\in(\mathbb{S}^1)^{\mathbb{Z}}$ and $ax=(a,x_1,x_2,\ldots)\in (\mathbb{S}^1)^{\mathbb{Z}}$ and $da$ is the normalized Lebesgue measure on $\mathbb{S}^1$. Let $\mu$ the Gibbs measure constructed from $\mathcal{L}_A$ in the standard way.

Let $\Lambda_n=[0,\ldots,n]\cap\mathbb{Z}$ and $\mathscr{T}_{\Lambda_n}$ the external $\sigma$-algebra as defined in the Georgii's book.

Question. Is there any Gibbs specification $\gamma=(\gamma_{\Lambda_n})_{n\in\mathbb{N}}$ such that

$$ \mu(A|\mathscr{T}_{\Lambda_n})(x)=\gamma(A|x) \quad \mu-\text{a.s.} ? $$

The answer is simple if $A$ is a potential depending only on finite number of coordinates or other words, if $A$ is short range potential. Since I am considering $A$ lipschtiz it seems reasonable to fix some configuration $\omega_0\in\mathbb{S}^{\mathbb{N}}$ and "project" the potential with this boundary condition considering potentials (in the sense of statistical mechanics)

$$ \Phi_n=(\Phi^n_{\Gamma})_{\Gamma\subset \mathbb{N}} $$ given by $$ \Phi^n_{\Gamma}(x)= \left\{ \begin{array}{rl} -A(x_1,\ldots,x_{n},\omega_{n+1},\ldots),& \text{if}\ \Gamma=\{1,\ldots,n\}; \\ 0,&\text{otherwise}. \end{array} \right. $$ and its respective specifications and try to prove that the unique $\mu_n\in\mathcal{G}(\Phi_n)$ converge to the measure $\mu$ for any choice of $\omega_0$.

I decided to post this question to know if there is a standard procedure to obtain the measure $\mu$ from the specification point of view or if there is an obvious reason for this construction, I describe above, do not give me a convergent sequence of measures. Any help or reference is welcome. Thanks.