Extremal infinite-volume Ising measures (${\mu }^0,{\mu }^1$)

For the ferromagnetic binary Ising model, all-zero boundary measures increase and all-one boundary measures decrease as boxes exhaust the lattice, in stochastic domination. The nearest-neighbor spatial conditional probability distribution reduces each domain comparison to a mixture of boundary conditions for the smaller box. Probabilities that a specified finite set is all one consequently converge; inclusion-exclusion gives convergence of every finite Ising spin pattern. Their consistent limits define the two extreme infinite-volume measures. Existence does not assert that the two limits coincide.