Construction of a specification from its singleton part

Abstract

We state a construction theorem for specications starting from single-site conditional probabilities (singleton part). We consider general single-site spaces and kernels that are absolutely continuous with respect to a chosen product mea- sure (free measure). Under a natural order-consistency assumption and weak non- nullness requirements we show existence and uniqueness of the specication ex- tending the given singleton part. We determine conditions granting the continuity of the specication. In addition, we show that, within a class of measures with suitable support properties, consistency with singletons implies consistency with the full specication.