  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.
