Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular we prove that every translation invariant Borel probability measure μ on the space X of finite-alphabet configurations on ℤ can be weakly approximated by Markov measures μ _n with supp(μ _n )=X and with the entropies h(μ _n )→h(μ). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.