Construction of a specification from its singleton part

We state a construction theorem for specifications 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 measure (free measure). Under a natural order-consistency assumption and weak non-nullness requirements we show existence and uniqueness of the specification extending the given singleton part. We determine conditions granting the continuity of the specification. In addition, we show that, within a class of measures with suitable support properties, consistency with singletons implies consistency with the full specification.