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Take upward-closed set families of subsets of $[n]$ of size at most $\ell$, closed under the forcing rule in Razborov closure. Normalize any family containing a set of size at most one to the full family, since its clique predicate is identically true. Intersection is the meet in a lattice; closure of the union is the join in a lattice. A family represents the graphs containing a clique on at least one of its members. The approximate meet can lose positive inputs and the approximate join can acquire negative inputs. The Razborov gate-by-gate approximation lemma charges output errors to those local errors.

Order configurations coordinatewise: $\omega \le \eta$ when $\omega (e)\le \eta (e)$ for every coordinate. An increasing event $A$ is an upward-closed set: $\omega \in A$ and $\omega \le \eta$ imply $\eta \in A$.

The finite-lattice FKG inequality states that a probability mass function satisfying the FKG lattice conditionis positively associated. Here meet in a lattice and join in a lattice are coordinatewise minimum and maximum. Thus for increasing observables $f,g$,Taking indicator functions gives $\mu (A\cap B)\ge \mu (A)\mu (B)$ for all increasing events. The strictly positive version suffices for the Ising weights here; the theorem also holds for nonnegative log-supermodular weights of positive total mass.

Work edge by edge. Let $\omega ,\eta$ have two-bit restrictions to an edge. If these restrictions are comparable, taking their meet in a lattice and join in a lattice only reorders the two pairs, so their total equality indicators do not change. If they are incomparable, they are $01$ and $10$, while their meet in a lattice and join in a lattice are $00$ and $11$. In that case the sum of equality indicators increases from zero to two. ConsequentlyThe partition functions cancel from the required ratio, leavingThis is the ferromagnetic Ising lattice inequality in binary-spin normalization, so the measure satisfies the FKG lattice condition and is positively associated. If $\sigma (x)=2\omega (x)-1$, then ${\delta }_{\omega (x),\omega (y)}=(1+\sigma (x)\sigma (y))/2$: the conventional $\{-1,1\}$ coupling is $\beta /2$, which explains the factor two in the edge calculation.

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 $\omega (z)$ givesBoth 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 $\Lambda \subset {\Lambda }^{\prime }$ 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 yieldsThese 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 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.