改进的齐次网格随机场的逼近定理

Approximation theorems of modified homogeneous lattice random fields

引入了测度v的熵h(v)和热力学形式体系中的一些概念,改进了齐次网格随机场理论中几个著名的逼近定理,得到了两个新的定理.这两个新的定理非常接近著名的网格随机场理论中的逼近定理,其逼近结果可以应用到信息论与维数理论中.证明了Zd,d≥1中有限格局空间X上的每一个转移不变Borel概率测度μ可由Markov测度μn来弱逼近,其中μn满足supp(μn)=X且熵h(μn)→h(μ).其证明是基于热力学形式体系中的一些事实.

The entropy h（υ） of measure υ and some notions from Thermodynamic Formalism are introduced. We refine some well-known approximation theorems in the theory of homogeneous lattice random fields and derive two new theorems. These are close to the well-known approximation theorems in the theory of homogeneous lattice random fields. The approximation results can apply to information theory and dimension theory. We prove that every translation invariant Borel probability measure on the space X of finite-alphabet configurations on Z^d ,d…1, can be weakly approximated by Markov measures μn with supp（μn）=X and with the entropies h（μn）→h（μ）. The proof is based on some facts of Thermodynamic Formalism.