Bayesian information topologies

A state of the world [ohgr] is drawn to some known prior distribution, [mgr]. Agents receive their information about [ohgr] and form their posterior distributions, P[ohgr], by conditioning on the sub-σ-fields that represent their information. The mapping [ohgr][map]P[ohgr] contains all of the economically relevant aspects of an agent’s information. ‘Bayesian’ topologies on these mappings are defined by saying that information structures are close if there is only small probability that the posterior distributions are far apart. This approach leads to a characterization of the relation between partition representations of information and σ-field representations of information.