Formalization of Information: Knowledge and Belief

• Abstract

Billingsley (1995) and Dubra-Echenique (2004) provide an example to show that the formalization of information by sigma-algebras and by partitions need not be equivalent. Although Herves-Beloso-Monteiro (2012) provide a method to generate a sigma-algebra from a partition and another method for going in the opposite direction, we show that their two methods are in fact based on two different notions of information: (i) information as belief, (ii) information as knowledge. We go on to two mutually exclusive cases. If information is conceived to allow for falsehoods, case (i) above, the equivalence between sigma-algebras and partitions holds after applying the notion of posterior-completion suggested by Brandenburger-Dekel (1987). If information is conceived not to allow for falsehoods, case (ii) above, the equivalence holds only for measurable partitions and countably-generated sigma-algebras.