A Compact Topology for Sigma-Algebra Convergence - Patrick Beissner; Jonas Tölle (Ludwig-Maximilians-Universität München und Humboldt-Universität zu Berlin, Collaborative Research Center Transregio 190 - Rationality and Competition, 2018)
We propose a sequential topology on the collection of sub-sigma-algebras included in a separable probability space. We prove compactness of the conditional expectations with respect to L2-bounded random variables along sequences of sub-sigma-algebras. The varying index of measurability is captured by a bundle space construction. As a consequence, we establish the compactness of the space of sub-sigma-algebras. The proposed topology preserves independence and is compatible with join and meet operations. Finally, a new application to information economics is discussed.
Keywords: 330; convergence of sigma-algebras; compactness of sub-sigma-algebras; conditional expectation; fiber bundle; information economics