Formalization of information: knowledge and belief

Billingsley (Probability and measure, Wiley, New Jersey, 1995) and Dubra and Echenique (Math Soc Sci 47(2):177–185, 2004) provide an example to show that the formalization of information by σ-algebras and by partitions need not be equivalent. Although Hervés-Beloso and Monteiro (Econ Theory 54(2):405–418, 2013) provide a method to generate a σ-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. If information is conceived to allow for falsehoods, case (i) above, the equivalence between σ-algebras and partitions holds after applying the notion of posterior completion suggested by Brandenburger and Dekel (J Math Econ 16(3):237–245, 1987). If information is conceived not to allow for falsehoods, case (ii) above, the equivalence holds only for measurable partitions and countably generated σ-algebras.