= Solution
With the inner boundary specified in the PDF, the <Ising boundary condition> pins $\omega|_{\partial\Lambda}=\zeta$. Explicitly $\mu_\Lambda^\zeta$ is $\mu_\Lambda$ conditioned on those <Ising spins>. Its <partition function> sums over unpinned interior <Ising spins> only; <edges> between two pinned sites contribute a constant that cancels on normalization.
Pinning coordinates preserves the <FKG lattice condition>, since <meet in a lattice> and <join in a lattice> still have the same pinned values. It also gives monotonicity in the pins. For any <increasing observable> $f$, <positive association of random variables> with an unpinned <Ising spin> $\omega(z)$ gives
$$
E[f\mid\omega(z)=1]\ge E[f\mid\omega(z)=0].
$$
Both <conditional probabilities> are positive. The same inequality holds after other <Ising spins> have been pinned. Changing pins one at a time proves <stochastic domination> of the measures as $\zeta$ increases.
Now let $\Lambda\subset\Lambda'$ and take an <increasing observable> depending on a fixed set inside the smaller box. Given the <Ising spins> on $\partial\Lambda$, the larger-box measure induces precisely the smaller-box interior law with those boundary values: nearest-neighbor interactions cannot see farther once this boundary is fixed. Its conditional law is therefore between the all-zero and all-one smaller-box laws. Averaging yields
$$
\mu_\Lambda^0(f)\le\mu_{\Lambda'}^0(f),\qquad
\mu_{\Lambda'}^1(f)\le\mu_\Lambda^1(f).
$$
These are bounded monotone sequences along any increasing box exhaustion.
For every finite set $S$, the event that all <Ising spins> in $S$ are one is increasing, so its <probability> converges in both boundary sequences. <Probabilities> of arbitrary finite zero/one patterns are finite <inclusion-exclusion> combinations of these all-one <probabilities>, and hence converge too. The limits are consistent <finite-dimensional distributions>; they define measures on $\{0,1\}^{\mathbb Z^d}$ and give <weak convergence of probability measures> in the <product topology>. Thus \b[both <extremal infinite-volume Ising measures> $\mu^0,\mu^1$ exist], with $\mu^0$ below $\mu^1$ in <stochastic domination>. This argument proves existence even when the two limits are different.
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