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. This paper provides a Bayesian definition of common knowledge, that is, a definition in terms of beliefs (probability measures). The main result is an equivalence between this definition and a definition in terms of the σ-fields representing 1 and 2's information. To obtain this result the conditional probabilities must be proper and the σ-fields posterior completed.