Description of random fields by means of one-point finite-conditional distributions

The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ℤ^ν and the conditional probability measures at one point given the values on a finite subset of the lattice ℤ^ν. We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.