Let $Omega_{0}:={-1,1}$, $mathcal{F}_{0} = 2^{Omega_{0}}$ and $nu_{0}$ a given measure on $(Omega_{0},mathcal{F}_{0})$. If $Lambda subset mathbb{Z}^{d}$ is finite, we take $Omega_{Lambda} := Omega_{0}^{Lambda}$, $mathcal{F}_{Lambda}$ the $|Lambda|$-fold product $sigma$-algebra of $mathcal{F}_{0}$ with iself and $nu_{Lambda}$ the corresponding product measure. An element $omega=(omega_{x})_{xin Lambda}in Omega_{Lambda}$ is called a configuration of the system. Consider the (nearest neighborhood) Hamiltonian with free Boundary conditions and field $h in mathbb{R}$, which is a function $H_{Lambda,h}^{emptyset}:Omega_{Lambda} to mathbb{R}$ defined by:

begin{eqnarray}

H_{Lambda, beta}^{emptyset}(omega) := -sum_{substack{x,y in Lambda \xsim y}}omega_{x}omega_{y} – hsum_{xin Lambda}omega_{x} tag{1}label{1}

end{eqnarray}

The finite volume Gibbs state with free boundary conditions is the probability measure $mu_{Lambda, beta,h}^{emptyset}$ given by

begin{eqnarray}

dmu_{Lambda,beta,h}^{emptyset}(omega) = frac{1}{Z_{Lambda,beta, h}^{emptyset}}e^{-beta H_{Lambda, h}^{emptyset}(omega)}dnu_{Lambda}(omega) tag{2}label{2}

end{eqnarray}

where the partition function $Z_{Lambda,beta,h}^{emptyset}$ normalizes the measure. Now, take $Omega := Omega_{0}^{mathbb{Z}^{d}}$, $mathcal{F}$ the (infinite) product $sigma$-algebra on $Omega$ and $nu$ the corresponding product measure.If $eta in Omega$, it is convenient to define $Omega_{Lambda}^{eta}:={omega in Omega: hspace{0.1cm} mbox{$omega_{x} = eta_{x}$ if $x in Lambda^{c}$}}$. The Hamiltonian with $eta$ as a boundary condition is a function on $Omega_{Lambda}^{eta}$ given by:

begin{eqnarray}

H_{Lambda,beta}^{eta}(omega) := -sum_{substack{{x,y}cap Lambda neq emptyset \ xsim y}}omega_{x}omega_{y}-hsum_{xin Lambda}omega_{x} tag{3}label{3}

end{eqnarray}

The finite volume Gibbs state, in this case, is the probability measure on $Omega_{Lambda}^{eta}$ (with discrete $sigma$-algebra) given by:

begin{eqnarray}

dmu_{Lambda,beta,h}^{eta}(omega) = frac{1}{Z_{Lambda,beta, h}^{eta}}e^{-beta H_{Lambda, h}^{eta}(omega)}dnu_{Lambda}(omega) tag{4}label{4}

end{eqnarray}

To study weak convergence of Gibbs states one would like to extend the above Gibbs states to a Gibbs state on all $Omega$. For the case of $eta$ boundary conditions, this can be done by defining on $(Omega, mathcal{F})$ a new probability measure by (with abuse of notation):

begin{eqnarray}

dmu_{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)}dnu_{Lambda}(omega) quad mbox{if $omega_{x} = eta_{x}$ for all $xin Lambda^{c}$} \

displaystyle 0 quad mbox{otherwise}

end{cases}

tag{5}label{5}

end{eqnarray}

where $Pi_{Lambda, eta}$ is the canonical projection $Omega to Omega_{Lambda}^{eta}$. Now, (ref{5}) cannot extend (ref{2}) because we cannot take $eta$ to be zero outise some $Lambda$, since $Omega_{0} ={-1,1}$.

**Question:** How to extend the finite volume Gibbs measure to $Omega$?

**Possible solution:** If I’m not mistaken, $mu_{Lambda, beta, h}^{eta}$ on $Omega$ as define by (ref{5}) is just the pushfoward of the corresponding measure $mu_{Lambda,beta,h}^{eta}$ defined by (ref{4}) with respect to the canonical injection $i_{eta}:Omega_{Lambda}^{eta}hookrightarrow Omega$. So I’m guessing $mu_{Lambda,beta,h}^{emptyset}$ can be extended to $Omega$ in the same way by taking the pushfoward of $mu_{Lambda,beta,h}^{emptyset}$ given by (ref{2}) with respect to the injection $i_{0}:Omega_{Lambda}hookrightarrow Omega$?