Agreeing to agree

Aumann (1976) shows that agents who have a common prior cannot have common knowledge of their posteriors for event E if these posteriors do not coincide. But given an event E, can the agents have posteriors with a common prior such that it is common knowledge that the posteriors for Edo coincide? We show that a necessary and sufficient condition for this is the existence of a nonempty finite event F with the following two properties. First, it is common knowledge at F that the agents cannot tell whether E occurred. Second, this still holds true at F, when F itself becomes common knowledge.