Common Knowledge with Probability 1

Two people, 1 and 2, are said to have common knowledge of an event if both know it, 1 knows that 2 knows it, 2 knows that 1 knows it, 1 knows that 2 knows that 1 knows it, and so on. We provide a Bayesian definition of common knowledge, that is, a definition in terms of beliefs (probability measures). Our main result is an equivalence between this definition and a definition in terms of the a-fields representing 1 and 2s information. To obtain this result we need the conditional probabilities to be proper and the a-fields to be posterior completed.